One of the strings of a guitar lies along the x-axis when in equilibrium. The en

Question

One of the strings of a guitar lies along the x-axis when in equilibrium. The end of the string at x=0 (the bridge of the guitar) is tied down. An incident sinusoidal wave travels along the string in the x-direction at 143 m/s with an amplitude of 0.750 mm and a frequency of 440 Hz. This wave is reflected from the fed end at x=0, and the reflected traveling wave forms a standing wave. a) Find the equation giving the displacement of a point in the string as a function of position and time. b) Locate the point on the string that do not move at all. c) Find the amplitude, maximum transverse velocity, and maximum transverse acceleration at points of maximum oscillation.

Explanation / Answer

For the calculation of the velocity of the given wave the equation of the wave is more than enough.

We, know a wave travelling across a string i +ve X direction is given by,

y=Asin(kx-wt)
where,
y is displacement
A is amplitude
k is wave number or propagation constant or simply a constant for given wave
w is angular velocity

Now, rewriting given eq. of wave
y=15sin((pi/6*x) -10pi*t)

Now comparing this eq. with standard eq. we get,
A=15,
k=pi/6
w=10pi

Now, we know
k=2pi/L, where, L is wavelength,
=> L=2pi/k

velocity, v=f*L where, f is frequency
or,v=2pi*f/k
or, v=w/k
Now,using above relation we get
v=10pi/(pi/6)
=>v=60m/s

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