A transverse sinusoidal wave is moving along a string in the positive direction

Question

A transverse sinusoidal wave is moving along a string in the positive direction of an x axis with a speed of 80 m/s. At t = 0, the string particle at x = 0 has a transverse displacement of 4.4 cm from its equilibrium position and is not moving. The maximum transverse speed of the string particle at x = 0 is 20 m/s. (a) What is the frequency of the wave? (b) What is the wavelength of the wave? If the wave equation is of the form y(x, t) = Ym sin(kx + - omega t + Phi), what are (c) Ym, (d) k, (e) omega, (f) Phi, and (g) the correct choice of sign in front of omega?

Explanation / Answer

1)
y(x, t) = ym*sin(kx +/- t + )
dy/dt = +/-*ym*cos(kx +/- t + )
Since the wave is moving along the positive x direction, we can take the negative sign infront of .

At x = 0, t = 0 -

4.4 * 10^-2 = ym*sin()
ym = (4.4 * 10^-2) / sin() -----------1


dy/dt(0,0 ) = 0 = -*ym*cos()
Therefore, = 90.

Thus, ym = (4.4 * 10^-2) / sin()
ym = 4.4 * 10^-2 m.

The maximum value of dy/dt(0,t):
20 m/s = *ym
= 20/( 4.4 * 10^-2 ) = 454.55 rad/s

a)
f = w/(2*pi) = 72.4 (Hz)
Frequency of the wave, f = 72.4 hz

b)
Wave Length = 2*pi/k = 2*pi/(/c)
Wave Length = 2*pi*c/
Wave Length = 2*pi*80/454.55
Wave Length = 1.11 m

c) ym = 4.4 * 10^-2 (m)
d)
k = /c = 454.55/80
k = 5.68
e) = 454.55 rad/sec

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