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: (c) $y = 3 \sin(5x - 0.5t) + 4 \cos(5x - 0.5t)$

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 $s^{-1}$).

A transverse harmonic wave on a string is described by $y(x, t) = 3.0 \sin (36 t + 0.018 x + \pi/4)$ 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 ?

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 $s^{-1}$

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.

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?

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: (d) $y = \cos x \sin t + \cos 2x \sin 2t$

What is the amplitude of a point 0.375 m away from one end?

Use the formula $v = \sqrt{\frac{\gamma P}{\rho}}$ to explain why the speed of sound in air (b) increases with temperature,

A transverse harmonic wave on a string is described by $y(x, t) = 3.0 \sin (36 t + 0.018 x + \pi/4)$ where $x$ and $y$ are in cm and $t$ in s. The positive direction of $x$ is from left to right. (b) What are its amplitude and frequency ?

Use the formula $v = \sqrt{\frac{\gamma P}{\rho}}$ to explain why the speed of sound in air (a) is independent of pressure,

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) $y = 2 \cos(3x) \sin(10t)$

The transverse displacement of a string (clamped at its both ends) is given by $y(x, t) = 0.06 \sin(\frac{2\pi x}{3}) \cos(120 \pi t)$ 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 : (c) Determine the tension in the string.

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?

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 (b) the tension in the string?

Use the formula $v = \sqrt{\frac{\gamma P}{\rho}}$ to explain why the speed of sound in air (c) increases with humidity.

The transverse displacement of a string (clamped at its both ends) is given by $y(x, t) = 0.06 \sin(\frac{2\pi x}{3}) \cos(120 \pi t)$ 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 : (b) Interpret the wave as a superposition of two waves travelling in opposite directions. What is the wavelength, frequency, and speed of each wave ?

For the wave on a string described in Exercise 15.11, do all the points on the string oscillate with the same (c) amplitude? Explain your answers.

Explain why (or how): (a) in a sound wave, a displacement node is a pressure antinode and vice versa,

For the wave on a string described in Exercise 15.11, do all the points on the string oscillate with the same (a) frequency? Explain your answers.