Missing data: the wave number
![k=60 cm^{-1}](https://tex.z-dn.net/?f=k%3D60%20cm%5E%7B-1%7D)
(a) ![z = 0.003 sin (6000y-31.4 t)](https://tex.z-dn.net/?f=z%20%3D%200.003%20sin%20%286000y-31.4%20t%29)
For a transverse wave travelling in the positive y-direction and with vibration along the z-direction, the equation of the wave is
![z = A sin (ky-\omega t)](https://tex.z-dn.net/?f=z%20%3D%20A%20sin%20%28ky-%5Comega%20t%29)
where
A is the amplitude of the wave
k is the wave number
is the angular frequency
t is the time
In this situation:
A = 3.0 mm = 0.003 m is the amplitude
is the wave number
is the period, so the angular frequency is
![\omega=\frac{2\pi}{T}=\frac{2\pi}{0.20}=31.4 rad/s](https://tex.z-dn.net/?f=%5Comega%3D%5Cfrac%7B2%5Cpi%7D%7BT%7D%3D%5Cfrac%7B2%5Cpi%7D%7B0.20%7D%3D31.4%20rad%2Fs)
So, the wave equation (in meters) is
![z = 0.003 sin (6000y-31.4 t)](https://tex.z-dn.net/?f=z%20%3D%200.003%20sin%20%286000y-31.4%20t%29)
(b) 0.094 m/s
For a transverse wave, the transverse speed is equal to the derivative of the displacement of the wave, so in this case:
![v_t = z' = -A \omega cos (ky-\omega t)](https://tex.z-dn.net/?f=v_t%20%3D%20z%27%20%3D%20-A%20%5Comega%20cos%20%28ky-%5Comega%20t%29)
So the maximum transverse wave occurs when the cosine term is equal to 1, therefore the maximum transverse speed must be
![v_{t}_{max} =\omega A](https://tex.z-dn.net/?f=v_%7Bt%7D_%7Bmax%7D%20%3D%5Comega%20A)
where
![\omega = 31.4 rad/s\\A = 0.003 m](https://tex.z-dn.net/?f=%5Comega%20%3D%2031.4%20rad%2Fs%5C%5CA%20%3D%200.003%20m)
Substituting,
![v_{t}_{max}=(31.4)(0.003)=0.094 m/s](https://tex.z-dn.net/?f=v_%7Bt%7D_%7Bmax%7D%3D%2831.4%29%280.003%29%3D0.094%20m%2Fs)
(c) 5.24 mm/s
The wave speed is given by
![v=f \lambda](https://tex.z-dn.net/?f=v%3Df%20%5Clambda)
where
f is the frequency of the wave
is the wavelength
The frequency can be found from the angular frequency:
![f=\frac{\omega}{2\pi}=\frac{31.4}{2\pi}=5 Hz](https://tex.z-dn.net/?f=f%3D%5Cfrac%7B%5Comega%7D%7B2%5Cpi%7D%3D%5Cfrac%7B31.4%7D%7B2%5Cpi%7D%3D5%20Hz)
While the wavelength can be found from the wave number:
![\lambda = \frac{2\pi}{k}=\frac{2\pi}{6000}=1.05\cdot 10^{-3} m](https://tex.z-dn.net/?f=%5Clambda%20%3D%20%5Cfrac%7B2%5Cpi%7D%7Bk%7D%3D%5Cfrac%7B2%5Cpi%7D%7B6000%7D%3D1.05%5Ccdot%2010%5E%7B-3%7D%20m)
Therefore, the wave speed is
![v=(5)(1.05\cdot 10^{-3} )=5.24 \cdot 10^{-3} m/s = 5.24 mm/s](https://tex.z-dn.net/?f=v%3D%285%29%281.05%5Ccdot%2010%5E%7B-3%7D%20%29%3D5.24%20%5Ccdot%2010%5E%7B-3%7D%20m%2Fs%20%3D%205.24%20mm%2Fs)