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Chapter 15 Waves
Introduction
A wave is a pattern of disturbance that propagates from one point to another, transporting energy and information, without the net transfer of matter as a whole. When a pebble is dropped into a pond, the disturbance travels outward in circles, but the water itself only moves up and down. Similarly, sound travels through the air as a disturbance, not as a bulk flow of air.
Our ability to communicate, from speech to modern telecommunications, relies on the transmission of signals through waves.
Types of Waves
Waves can be broadly classified into three categories:
- Mechanical Waves: These waves require a material medium for their propagation. They cannot travel through a vacuum. Their propagation depends on the elastic and inertial properties of the medium. Examples include waves on a string, water waves, and sound waves.
- Electromagnetic Waves: These waves do not require a material medium and can travel through a vacuum. They consist of oscillating electric and magnetic fields. Examples include light, radio waves, and X-rays. In a vacuum, all electromagnetic waves travel at the speed of light, $c \approx 3 \times 10^8 \text{ m/s}$.
- Matter Waves: These waves are associated with moving particles like electrons, protons, and atoms, as described by quantum mechanics. They are fundamental to modern technologies like the electron microscope.
This chapter will focus on mechanical waves.
Waves and Oscillations
The propagation of mechanical waves in an elastic medium is intimately connected to the concept of oscillations. A medium can be visualized as a collection of coupled oscillators (like a series of connected springs or atoms in a crystal lattice). When one part of the medium is disturbed, its oscillation is transferred to the next part due to the elastic forces, causing the disturbance to propagate as a wave.
Transverse and Longitudinal Waves
Mechanical waves are classified based on the direction of oscillation of the medium's particles relative to the direction of wave propagation.
Transverse Waves
A transverse wave is a wave in which the particles of the medium oscillate perpendicular (transverse) to the direction of wave propagation.
- Example: Waves on a stretched string. If the string is shaken up and down, the wave travels horizontally, but the string segments move vertically.
- Transverse waves involve shearing strain. Therefore, they can only be propagated in media that can sustain shearing stress, which are primarily solids. They cannot propagate through ideal fluids (liquids and gases).
Longitudinal Waves
A longitudinal wave is a wave in which the particles of the medium oscillate parallel to the direction of wave propagation.
- Example: Sound waves in air. The wave travels as a series of compressions (regions of high density and pressure) and rarefactions (regions of low density and pressure), with air molecules oscillating back and forth along the direction of sound travel.
- Longitudinal waves involve compressive strain. Therefore, they can propagate in any elastic medium: solids, liquids, and gases.
Note: Water waves on the surface of deep water are a combination of both transverse and longitudinal motions; the water particles move in roughly circular paths.
Displacement Relation in a Progressive Wave
A traveling or progressive wave can be described by a mathematical function of both position ($x$) and time ($t$). For a sinusoidal wave traveling in the positive x-direction, this function is:
$y(x, t) = A \sin(kx - \omega t + \phi)$
For a wave traveling in the negative x-direction, the sign of the $\omega t$ term is changed:
$y(x, t) = A \sin(kx + \omega t + \phi)$
Here, $y$ represents the transverse displacement. For a longitudinal wave, it would be replaced by a variable like $s$ for displacement along the direction of propagation.
Key Wave Parameters
- Amplitude (A): The maximum displacement of a particle from its equilibrium position. It is always a positive quantity.
- Phase ($kx - \omega t + \phi$): The argument of the sine or cosine function, which determines the displacement of the wave at any position and time.
- Initial Phase or Phase Constant ($\phi$): The phase of the wave at $x=0$ and $t=0$. It determines the initial state of the wave.
- Wavelength ($\lambda$): The minimum spatial distance between two points on the wave that are in the same phase (e.g., the distance between two consecutive crests or troughs). Its SI unit is the metre (m).
- Angular Wave Number (k): Related to the wavelength by:
- Period (T): The time taken for one complete oscillation of a particle of the medium. Its SI unit is the second (s).
- Frequency ($\nu$): The number of oscillations per unit time. $\nu = 1/T$. Its SI unit is the hertz (Hz).
- Angular Frequency ($\omega$): Related to the period and frequency by:
$k = \frac{2\pi}{\lambda}$
It represents the spatial frequency of the wave (how rapidly the wave oscillates in space). Its SI unit is rad m⁻¹.
$\omega = \frac{2\pi}{T} = 2\pi\nu$
It represents the temporal frequency of the wave (how rapidly the wave oscillates in time). Its SI unit is rad s⁻¹.
Example 1. A wave travelling along a string is described by $y(x, t) = 0.005 \sin(80.0x - 3.0t)$, where the numerical constants are in SI units. Calculate (a) the amplitude, (b) the wavelength, and (c) the period and frequency of the wave. Also, calculate the displacement y at $x = 30.0$ cm and $t = 20$ s.
Answer:
Comparing the given equation with the standard form $y(x, t) = A \sin(kx - \omega t)$, we identify the parameters:
(a) Amplitude (A):
$A = 0.005$ m.
(b) Wavelength ($\lambda$):
The angular wave number is $k = 80.0 \text{ rad m}^{-1}$.
$\lambda = \frac{2\pi}{k} = \frac{2\pi}{80.0} \approx 0.0785 \text{ m} = 7.85 \text{ cm}$.
(c) Period (T) and Frequency ($\nu$):
The angular frequency is $\omega = 3.0 \text{ rad s}^{-1}$.
$T = \frac{2\pi}{\omega} = \frac{2\pi}{3.0} \approx 2.09 \text{ s}$.
$\nu = \frac{1}{T} = \frac{1}{2.09} \approx 0.48 \text{ Hz}$.
Displacement at $x=0.3$ m and $t=20$ s:
$y(0.3, 20) = 0.005 \sin(80.0 \times 0.3 - 3.0 \times 20)$
$y = 0.005 \sin(24 - 60) = 0.005 \sin(-36 \text{ rad})$
To evaluate $\sin(-36)$, we can use the periodicity of sine. Note that $36 \text{ rad} \approx 11.46\pi$. We can write $\sin(-36) = -\sin(36) = -\sin(36 - 12\pi) = -\sin(36 - 37.7) = -\sin(-1.7) = \sin(1.7 \text{ rad})$.
$y \approx 0.005 \sin(1.7) \approx 0.005 \times 0.9917 \approx 0.00496 \text{ m} \approx 5 \text{ mm}$.
The Speed of a Travelling Wave
The speed of a wave ($v$) is the speed at which the wave pattern (e.g., a crest) propagates through the medium. It is related to the wavelength and period.
Derivation of Wave Speed
A point of constant phase on a traveling wave moves such that the phase itself, $kx - \omega t = \text{constant}$.
Differentiating this expression with respect to time:
$k \frac{dx}{dt} - \omega = 0$
Since the speed of the point of constant phase is $v = dx/dt$, we get:
$v = \frac{\omega}{k}$
Substituting $\omega = 2\pi/T$ and $k = 2\pi/\lambda$, we get the fundamental wave speed relation:
$v = \frac{2\pi/T}{2\pi/\lambda} = \frac{\lambda}{T} = \nu\lambda$
The speed of a mechanical wave is determined by the properties of the medium, specifically its elastic properties (which provide the restoring force) and its inertial properties (which resist acceleration).
Speed of a Transverse Wave on a Stretched String
The speed of a transverse wave on a string depends on the tension in the string ($T$) and its mass per unit length (linear mass density, $\mu$). The formula is:
$v = \sqrt{\frac{T}{\mu}}$
A higher tension results in a faster wave, while a thicker (heavier) string results in a slower wave.
Speed of a Longitudinal Wave (Speed of Sound)
The speed of a longitudinal wave depends on the elastic modulus and the density of the medium.
In a fluid (liquid or gas), the speed is given by:
$v = \sqrt{\frac{B}{\rho}}$
where $B$ is the bulk modulus and $\rho$ is the density.
In a solid rod, the speed is given by:
$v = \sqrt{\frac{Y}{\rho}}$
where $Y$ is the Young's modulus.
Newton's Formula and Laplace's Correction for Sound in a Gas
Newton initially assumed that the compressions and rarefactions in a sound wave occur under isothermal (constant temperature) conditions. For an isothermal process, the bulk modulus of a gas is equal to its pressure, $B=P$. This led to Newton's formula:
$v = \sqrt{\frac{P}{\rho}}$
This formula gives a value for the speed of sound in air at STP that is about 15% too low.
Laplace later corrected this by pointing out that the compressions and rarefactions happen so quickly that there is no time for heat to flow, so the process is adiabatic, not isothermal. For an adiabatic process, the bulk modulus is $B_{ad} = \gamma P$, where $\gamma = C_p/C_v$ is the ratio of specific heats. This gives Laplace's corrected formula:
$v = \sqrt{\frac{\gamma P}{\rho}}$
This formula gives a value that agrees very well with the measured speed of sound.
The Principle of Superposition of Waves
When two or more waves travel through the same medium at the same time, the resultant displacement at any point is the algebraic sum of the displacements due to the individual waves.
$y(x, t) = y_1(x, t) + y_2(x, t) + \dots$
This principle applies to linear waves, where the medium's response is proportional to the disturbance.
Interference of Waves
The phenomenon of superposition is also known as interference. Let's consider two sinusoidal waves of the same amplitude $A$, frequency $\omega$, and wave number $k$, traveling in the same direction but with a phase difference of $\phi$.
$y_1(x, t) = A \sin(kx - \omega t)$
$y_2(x, t) = A \sin(kx - \omega t + \phi)$
The resultant wave is: $y(x, t) = y_1 + y_2 = \left[2A \cos\left(\frac{\phi}{2}\right)\right] \sin\left(kx - \omega t + \frac{\phi}{2}\right)$.
The resultant wave is also a sinusoidal wave with a new amplitude, $A_{res} = 2A \cos(\phi/2)$.
- Constructive Interference: When the waves are in phase ($\phi = 0, 2\pi, 4\pi, \dots$), $\cos(\phi/2) = 1$, and the resultant amplitude is maximum, $A_{res} = 2A$. The crests of one wave align with the crests of the other.
- Destructive Interference: When the waves are out of phase ($\phi = \pi, 3\pi, 5\pi, \dots$), $\cos(\phi/2) = 0$, and the resultant amplitude is zero, $A_{res} = 0$. The crests of one wave align with the troughs of the other, causing them to cancel out.
Reflection of Waves
When a traveling wave encounters a boundary between two different media, it is partially reflected and partially transmitted. The nature of the reflection depends on the properties of the boundary.
Reflection from a Rigid Boundary
When a wave reflects from a fixed or rigid end (e.g., a string tied to a wall), the reflected wave is inverted. This corresponds to a phase change of $\pi$ radians (180°).
This occurs because the boundary point cannot move (it must be a node), so the reflected wave must exactly cancel the incident wave at the boundary at all times.
If incident wave is $y_i(x, t) = A \sin(kx - \omega t)$, reflected wave is $y_r(x, t) = A \sin(kx + \omega t + \pi) = -A \sin(kx + \omega t)$.
Reflection from a Free (Open) Boundary
When a wave reflects from a free end (e.g., a string attached to a light, frictionless ring on a rod), the reflected wave is not inverted. There is no phase change (a phase change of 0 radians).
At a free end, the restoring force is zero, which corresponds to a point of maximum displacement (an antinode).
If incident wave is $y_i(x, t) = A \sin(kx - \omega t)$, reflected wave is $y_r(x, t) = A \sin(kx + \omega t)$.
Standing Waves and Normal Modes
When two identical waves travel in opposite directions in the same medium (such as an incident wave and its reflection), their superposition creates a standing wave or stationary wave.
The equation for a standing wave is:
$y(x, t) = [2A \sin(kx)] \cos(\omega t)$
Key properties of standing waves:
- The wave pattern does not travel; it oscillates in place.
- The amplitude, $2A \sin(kx)$, varies with position $x$.
- Nodes: Points of permanently zero amplitude (where $\sin(kx)=0$). They are located at $x = n\lambda/2$.
- Antinodes: Points of maximum amplitude ($2A$) (where $|\sin(kx)|=1$). They are located at $x = (n+1/2)\lambda/2$.
- The distance between two consecutive nodes (or antinodes) is $\lambda/2$.
Normal Modes
When a medium is confined between two boundaries (like a string fixed at both ends), standing waves can only be formed for specific wavelengths and frequencies that satisfy the boundary conditions. These allowed frequencies are called natural frequencies or normal modes of oscillation.
- String Fixed at Both Ends (Length L): Both ends must be nodes. This requires $L = n\frac{\lambda}{2}$, where $n=1, 2, 3, \dots$. The allowed frequencies are $\nu_n = n\left(\frac{v}{2L}\right)$. All harmonics (multiples of the fundamental frequency, $\nu_1$) are present.
- Pipe Closed at One End (Length L): The closed end is a node, and the open end is an antinode. This requires $L = (2n-1)\frac{\lambda}{4}$, where $n=1, 2, 3, \dots$. The allowed frequencies are $\nu_n = (2n-1)\left(\frac{v}{4L}\right)$. Only odd harmonics are present.
- Pipe Open at Both Ends (Length L): Both ends must be antinodes. This requires $L = n\frac{\lambda}{2}$, where $n=1, 2, 3, \dots$. The allowed frequencies are $\nu_n = n\left(\frac{v}{2L}\right)$. All harmonics are present.
Beats
Beats is the phenomenon of periodic variation in the loudness (intensity) of sound that occurs when two sound waves of slightly different frequencies are superposed. The sound intensity waxes and wanes periodically.
If two waves with frequencies $\nu_1$ and $\nu_2$ interfere, the resulting sound wave has a frequency equal to the average of the two frequencies, $\nu_{avg} = (\nu_1+\nu_2)/2$. The amplitude of this wave, however, is modulated, varying slowly with time. The frequency of this amplitude variation is called the beat frequency.
$\nu_{beat} = |\nu_1 - \nu_2|$
Each "beat" corresponds to one cycle of waxing and waning of the sound. This phenomenon is used by musicians to tune their instruments.
Example 2. Two sitar strings A and B playing the note ‘Dha’ are slightly out of tune and produce beats of frequency 5 Hz. The tension of string B is slightly increased, and the beat frequency is found to decrease to 3 Hz. What is the original frequency of B if the frequency of A is 427 Hz?
Answer:
Let the frequency of string A be $\nu_A = 427$ Hz and the original frequency of string B be $\nu_B$.
The beat frequency is $|\nu_A - \nu_B| = 5$ Hz. This gives two possibilities for the original frequency of B:
1. $\nu_B = \nu_A + 5 = 427 + 5 = 432$ Hz.
2. $\nu_B = \nu_A - 5 = 427 - 5 = 422$ Hz.
When the tension in string B is increased, its frequency $\nu_B$ must increase (since $v = \sqrt{T/\mu}$ and $\nu = v/\lambda$).
Let's check the two possibilities with the new beat frequency of 3 Hz.
1. If the original $\nu_B$ was 432 Hz, increasing it would make it even larger (e.g., 433 Hz). The new beat frequency would be $|427 - 433| = 6$ Hz. This contradicts the observation that the beat frequency decreased.
2. If the original $\nu_B$ was 422 Hz, increasing it would bring it closer to 427 Hz (e.g., to 424 Hz). The new beat frequency would be $|427 - 424| = 3$ Hz. This matches the observation.
Therefore, the original frequency of string B was 422 Hz.
Doppler Effect
The Doppler effect is the apparent change in the frequency of a wave as a result of relative motion between the source of the wave and the observer. It is a common experience with sound waves: the pitch of an ambulance siren sounds higher as it approaches you and lower as it moves away.
General Formula
The observed frequency $\nu'$ is related to the source frequency $\nu$ by the general formula:
$\nu' = \nu \left(\frac{v \pm v_o}{v \mp v_s}\right)$
where:
- $v$: speed of the wave in the medium (e.g., speed of sound).
- $v_o$: speed of the observer.
- $v_s$: speed of the source.
Sign Convention
The signs in the numerator and denominator depend on the directions of motion.
- Numerator (Observer's motion): Use the + sign if the observer is moving towards the source. Use the - sign if the observer is moving away from the source.
- Denominator (Source's motion): Use the - sign if the source is moving towards the observer. Use the + sign if the source is moving away from the observer.
In general, motions that decrease the distance between the source and observer lead to an increase in the observed frequency (higher pitch), and motions that increase the distance lead to a decrease in frequency (lower pitch).
Example 3. A rocket is moving at a speed of 200 m s⁻¹ towards a stationary target. While moving, it emits a wave of frequency 1000 Hz. Some of the sound reaching the target gets reflected back to the rocket as an echo. Calculate (1) the frequency of the sound as detected by the target, and (2) the frequency of the echo as detected by the rocket. (Speed of sound in air = 330 m s⁻¹).
Answer:
1. Frequency detected by the target:
Here, the source (rocket) is moving towards the stationary observer (target).
Source frequency $\nu = 1000$ Hz.
Source speed $v_s = 200$ m/s (towards observer).
Observer speed $v_o = 0$.
Speed of sound $v = 330$ m/s.
Using the Doppler formula (observer stationary, source moving towards):
$\nu' = \nu \left(\frac{v}{v - v_s}\right) = 1000 \left(\frac{330}{330 - 200}\right) = 1000 \left(\frac{330}{130}\right) \approx 2538 \text{ Hz}$.
2. Frequency of the echo detected by the rocket:
Now, the target acts as a source of the echo. It reflects the sound at the frequency it received, so the new source frequency is $\nu' = 2538$ Hz. This source is stationary ($v_s=0$).
The rocket is now the observer, moving towards the stationary source of the echo.
Observer speed $v_o = 200$ m/s (towards source).
Using the Doppler formula (observer moving towards, source stationary):
$\nu'' = \nu' \left(\frac{v + v_o}{v}\right) = 2538 \left(\frac{330 + 200}{330}\right) = 2538 \left(\frac{530}{330}\right) \approx 4079 \text{ Hz}$.
Exercises
Question 15.1. A string of mass 2.50 kg is under a tension of 200 N. The length of the stretched string is 20.0 m. If the transverse jerk is struck at one end of the string, how long does the disturbance take to reach the other end?
Answer:
Question 15.2. A stone dropped from the top of a tower of height 300 m splashes into the water of a pond near the base of the tower. When is the splash heard at the top given that the speed of sound in air is 340 m s$^{–1}$ ? (g = 9.8 m s$^{–2}$)
Answer:
Question 15.3. A steel wire has a length of 12.0 m and a mass of 2.10 kg. What should be the tension in the wire so that speed of a transverse wave on the wire equals the speed of sound in dry air at 20 °C = 343 m s$^{–1}$.
Answer:
Question 15.4. Use the formula $v = \sqrt{\frac{\gamma P}{\rho}}$ to explain why the speed of sound in air
(a) is independent of pressure,
(b) increases with temperature,
(c) increases with humidity.
Answer:
Question 15.5. You have learnt that a travelling wave in one dimension is represented by a function $y = f(x, t)$ where x and t must appear in the combination $x – v t$ or $x + v t$, i.e. $y = f (x \pm v t)$. Is the converse true? Examine if the following functions for y can possibly represent a travelling wave :
(a) $(x – vt)^2$
(b) $\log[(x + vt)/x_0]$
(c) $1/(x + vt)$
Answer:
Question 15.6. A bat emits ultrasonic sound of frequency 1000 kHz in air. If the sound meets a water surface, what is the wavelength of (a) the reflected sound, (b) the transmitted sound? Speed of sound in air is 340 m s$^{–1}$ and in water 1486 m s$^{–1}$.
Answer:
Question 15.7. A hospital uses an ultrasonic scanner to locate tumours in a tissue. What is the wavelength of sound in the tissue in which the speed of sound is 1.7 km s$^{–1}$ ? The operating frequency of the scanner is 4.2 MHz.
Answer:
Question 15.8. A transverse harmonic wave on a string is described by
$y(x, t) = 3.0 \sin(36t + 0.018x + \pi/4)$
where x and y are in cm and t in s. The positive direction of x is from left to right.
(a) Is this a travelling wave or a stationary wave ?
If it is travelling, what are the speed and direction of its propagation ?
(b) What are its amplitude and frequency ?
(c) What is the initial phase at the origin ?
(d) What is the least distance between two successive crests in the wave ?
Answer:
Question 15.9. For the wave described in Exercise 15.8, plot the displacement (y) versus (t) graphs for x = 0, 2 and 4 cm. What are the shapes of these graphs? In which aspects does the oscillatory motion in travelling wave differ from one point to another: amplitude, frequency or phase ?
Answer:
Question 15.10. For the travelling harmonic wave
$y(x, t) = 2.0 \cos 2\pi(10t – 0.0080x + 0.35)$
where x and y are in cm and t in s. Calculate the phase difference between oscillatory motion of two points separated by a distance of
(a) 4 m,
(b) 0.5 m,
(c) $\lambda/2$,
(d) $3\lambda/4$
Answer:
Question 15.11. The transverse displacement of a string (clamped at its both ends) is given by
$y(x, t) = 0.06 \sin\left(\frac{2\pi x}{3}\right) \cos(120\pi t)$
where x and y are in m and t in s. The length of the string is 1.5 m and its mass is $3.0 \times 10^{–2}$ kg.
Answer the following :
(a) Does the function represent a travelling wave or a stationary wave?
(b) Interpret the wave as a superposition of two waves travelling in opposite directions. What is the wavelength, frequency, and speed of each wave ?
(c) Determine the tension in the string.
Answer:
Question 15.12. (i) For the wave on a string described in Exercise 15.11, do all the points on the string oscillate with the same (a) frequency, (b) phase, (c) amplitude? Explain your answers. (ii) What is the amplitude of a point 0.375 m away from one end?
Answer:
Question 15.13. Given below are some functions of x and t to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent (i) a travelling wave, (ii) a stationary wave or (iii) none at all:
(a) $y = 2 \cos(3x) \sin(10t)$
(b) $y = 2\sqrt{x - vt}$
(c) $y = 3 \sin(5x – 0.5t) + 4 \cos(5x – 0.5t)$
(d) $y = \cos x \sin t + \cos 2x \sin 2t$
Answer:
Question 15.14. A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of 45 Hz. The mass of the wire is $3.5 \times 10^{–2}$ kg and its linear mass density is $4.0 \times 10^{–2}$ kg m$^{–1}$. What is (a) the speed of a transverse wave on the string, and (b) the tension in the string?
Answer:
Question 15.15. A metre-long tube open at one end, with a movable piston at the other end, shows resonance with a fixed frequency source (a tuning fork of frequency 340 Hz) when the tube length is 25.5 cm or 79.3 cm. Estimate the speed of sound in air at the temperature of the experiment. The edge effects may be neglected.
Answer:
Question 15.16. A steel rod 100 cm long is clamped at its middle. The fundamental frequency of longitudinal vibrations of the rod are given to be 2.53 kHz. What is the speed of sound in steel?
Answer:
Question 15.17. A pipe 20 cm long is closed at one end. Which harmonic mode of the pipe is resonantly excited by a 430 Hz source ? Will the same source be in resonance with the pipe if both ends are open? (speed of sound in air is 340 m s$^{–1}$).
Answer:
Question 15.18. Two sitar strings A and B playing the note ‘Ga’ are slightly out of tune and produce beats of frequency 6 Hz. The tension in the string A is slightly reduced and the beat frequency is found to reduce to 3 Hz. If the original frequency of A is 324 Hz, what is the frequency of B?
Answer:
Question 15.19. Explain why (or how):
(a) in a sound wave, a displacement node is a pressure antinode and vice versa,
(b) bats can ascertain distances, directions, nature, and sizes of the obstacles without any “eyes”,
(c) a violin note and sitar note may have the same frequency, yet we can distinguish between the two notes,
(d) solids can support both longitudinal and transverse waves, but only longitudinal waves can propagate in gases, and
(e) the shape of a pulse gets distorted during propagation in a dispersive medium.
Answer:
Question 15.20. A train, standing at the outer signal of a railway station blows a whistle of frequency 400 Hz in still air. (i) What is the frequency of the whistle for a platform observer when the train (a) approaches the platform with a speed of 10 m s$^{–1}$, (b) recedes from the platform with a speed of 10 m s$^{–1}$? (ii) What is the speed of sound in each case ? The speed of sound in still air can be taken as 340 m s$^{–1}$.
Answer:
Question 15.21. A train, standing in a station-yard, blows a whistle of frequency 400 Hz in still air. The wind starts blowing in the direction from the yard to the station with a speed of 10 m s$^{–1}$. What are the frequency, wavelength, and speed of sound for an observer standing on the station’s platform? Is the situation exactly identical to the case when the air is still and the observer runs towards the yard at a speed of 10 m s$^{–1}$? The speed of sound in still air can be taken as 340 m s$^{–1}$
Answer:
Question 15.22. A travelling harmonic wave on a string is described by
$y(x, t) = 7.5 \sin(0.0050x + 12t + \pi/4)$
(a)what are the displacement and velocity of oscillation of a point at x = 1 cm, and t = 1 s? Is this velocity equal to the velocity of wave propagation?
(b)Locate the points of the string which have the same transverse displacements and velocity as the x = 1 cm point at t = 2 s, 5 s and 11 s.
Answer:
Question 15.23. A narrow sound pulse (for example, a short pip by a whistle) is sent across a medium. (a) Does the pulse have a definite (i) frequency, (ii) wavelength, (iii) speed of propagation? (b) If the pulse rate is 1 after every 20 s, (that is the whistle is blown for a split of second after every 20 s), is the frequency of the note produced by the whistle equal to 1/20 or 0.05 Hz ?
Answer:
Question 15.24. One end of a long string of linear mass density $8.0 \times 10^{–3}$ kg m$^{–1}$ is connected to an electrically driven tuning fork of frequency 256 Hz. The other end passes over a pulley and is tied to a pan containing a mass of 90 kg. The pulley end absorbs all the incoming energy so that reflected waves at this end have negligible amplitude. At t = 0, the left end (fork end) of the string x = 0 has zero transverse displacement (y = 0) and is moving along positive y-direction. The amplitude of the wave is 5.0 cm. Write down the transverse displacement y as function of x and t that describes the wave on the string.
Answer:
Question 15.25. A SONAR system fixed in a submarine operates at a frequency 40.0 kHz. An enemy submarine moves towards the SONAR with a speed of 360 km h$^{–1}$. What is the frequency of sound reflected by the submarine ? Take the speed of sound in water to be 1450 m s$^{–1}$.
Answer:
Question 15.26. Earthquakes generate sound waves inside the earth. Unlike a gas, the earth can experience both transverse (S) and longitudinal (P) sound waves. Typically the speed of S wave is about 4.0 km s$^{–1}$, and that of P wave is 8.0 km s$^{–1}$. A seismograph records P and S waves from an earthquake. The first P wave arrives 4 min before the first S wave. Assuming the waves travel in straight line, at what distance does the earthquake occur ?
Answer:
Question 15.27. A bat is flitting about in a cave, navigating via ultrasonic beeps. Assume that the sound emission frequency of the bat is 40 kHz. During one fast swoop directly toward a flat wall surface, the bat is moving at 0.03 times the speed of sound in air. What frequency does the bat hear reflected off the wall ?
Answer: