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18. Waves & Sound
Wave Functions
6:49 minutes
Problem 15.27b
Textbook Question
Textbook Question(II) A transverse wave pulse travels to the right along a string with a speed v = 2.0 m/s. At the shape of the pulse is given by the function D = 0.45 cos (2.6x + 1.2),
where D and x are in meters. (b) Determine a formula for the wave pulse at any time t assuming there are no frictional losses.
Verified step by step guidance
1
Identify the given parameters and the general wave function. The given function is D = 0.45 cos(2.6x + 1.2) and the wave speed v = 2.0 m/s.
Recognize that the wave function represents a cosine wave moving to the right. For a wave moving to the right, the wave function can be expressed in the form D(x, t) = A cos(kx - \\omega t + \\phi), where A is the amplitude, k is the wave number, \\omega is the angular frequency, and \\phi is the phase constant.
From the given wave function, extract the wave number k = 2.6 rad/m and the phase constant \\phi = 1.2 rad. The amplitude A = 0.45 m is also given.
Use the relationship between wave speed v, wave number k, and angular frequency \\omega, which is v = \\omega/k. Solve for \\omega using \\omega = vk = 2.0 m/s * 2.6 rad/m.
Substitute the values of A, k, \\omega, and \\phi into the general wave function to get the wave function at any time t: D(x, t) = 0.45 cos(2.6x - \\omega t + 1.2), where \\omega is calculated from the previous step.
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Textbook Question
A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is y(x, t)=2.30mm cos[(16.98 rad/m^)x+(742 rad/s)t]. Being more practical, you measure the rope to have a length of 1.35 m and a mass of 0.00338 kg. You are then asked to determine the following: (d) wave speed; (e) direction the wave is traveling;
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