Wave Equation
Physics ⇒ Waves and Sound
Wave Equation starts at 9 and continues till grade 12.
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Describe the physical meaning of the term 'node' in the context of standing waves.
Explain the difference between a traveling wave and a standing wave in terms of the wave equation.
Explain why the wave equation is linear and discuss the significance of this property.
If the amplitude of a wave is doubled, what happens to the speed of the wave?
A sound wave in air has a wavelength of 0.5 m and a speed of 340 m/s. What is its frequency?
A string has a mass per unit length of 0.02 kg/m and is under a tension of 80 N. What is the speed of a wave on this string?
A string of length 2 m is fixed at both ends and vibrates in its fundamental mode. If the speed of the wave on the string is 100 m/s, what is the frequency of the fundamental mode?
A wave is described by \( y(x, t) = 0.2 \sin(3x + 6t) \). What is the direction of propagation of the wave?
Which of the following best describes a standing wave?
(1) A wave that moves in one direction only
(2) A wave that appears stationary, with nodes and antinodes
(3) A wave that increases in amplitude with time
(4) A wave that loses energy as it propagates
Which of the following boundary conditions is appropriate for a string fixed at both ends?
(1) Displacement is zero at both ends
(2) Slope is zero at both ends
(3) Displacement is maximum at both ends
(4) Slope is maximum at both ends
Which of the following is a solution to the wave equation?
(1) \( y(x, t) = A \sin(kx - \omega t) \)
(2) \( y(x, t) = A e^{kx + \omega t} \)
(3) \( y(x, t) = A x^2 t \)
(4) \( y(x, t) = A \cos(kx + \omega t) \)
Which of the following is the general form of the one-dimensional wave equation?
(1) \( \frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2} \)
(2) \( \frac{\partial y}{\partial x} = v \frac{\partial y}{\partial t} \)
(3) \( \frac{\partial^2 y}{\partial t^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial x^2} \)
(4) \( \frac{\partial y}{\partial t} = v \frac{\partial y}{\partial x} \)
Fill in the blank: The angular frequency \( \omega \) is related to the frequency f by \( \omega = 2\pi \underline{\hspace{2cm}} \).
Fill in the blank: The general solution to the wave equation is a function of \( x \pm \underline{\hspace{2cm}} \).
Fill in the blank: The speed of a wave on a stretched string is given by \( v = \sqrt{\frac{T}{\mu}} \), where T is the ______ and \( \mu \) is the mass per unit length.
Fill in the blank: The term \( \frac{\partial^2 y}{\partial x^2} \) in the wave equation represents the ______ of the wave with respect to position.
True or False: The wave equation can be derived from Newton's laws for a vibrating string.
True or False: The wave equation can be used to describe the propagation of sound in air.
True or False: The wave equation can be written in three dimensions as \( \nabla^2 y = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2} \).
True or False: The wave equation describes the propagation of both mechanical and electromagnetic waves.
