The expression for the equation of a wave is:
![y(x,t)= A \cos (kx + \omega t)](https://tex.z-dn.net/?f=y%28x%2Ct%29%3D%20A%20%5Ccos%20%28kx%20%2B%20%5Comega%20t%29)
(1)
where
A is the amplitude
![k= \frac{2 \pi}{ \lambda }](https://tex.z-dn.net/?f=k%3D%20%5Cfrac%7B2%20%5Cpi%7D%7B%20%5Clambda%20%7D%20)
is the wave number, with
![\lambda](https://tex.z-dn.net/?f=%5Clambda)
being the wavelength
x is the displacement
![\omega= \frac{2 \pi}{T}](https://tex.z-dn.net/?f=%5Comega%3D%20%20%5Cfrac%7B2%20%5Cpi%7D%7BT%7D%20)
is the angular frequency, with T being the period
t is the time
The equation of the wave in our problem is
![y(x,t)= 1.6 \cos (0.71 x + 36 t)](https://tex.z-dn.net/?f=y%28x%2Ct%29%3D%201.6%20%5Ccos%20%280.71%20x%20%2B%2036%20t%29)
(2)
where x and y are in cm and t is in seconds.
a) Amplitude:
if we compare (1) and (2), we immediately see that the amplitude of the wave is the factor before the cosine:
A=1.6 cm
b) Wavelength:
we can find the wavelength starting from the wave number. For the wave of the problem,
![k= \frac{2 \pi}{\lambda}=0.71 cm^{-1}](https://tex.z-dn.net/?f=k%3D%20%5Cfrac%7B2%20%5Cpi%7D%7B%5Clambda%7D%3D0.71%20cm%5E%7B-1%7D%20)
And re-arranging this relationship we find
![\lambda= \frac{2 \pi}{k}= \frac{2 \pi}{0.71 cm^{-1}}=8.8 cm](https://tex.z-dn.net/?f=%5Clambda%3D%20%5Cfrac%7B2%20%5Cpi%7D%7Bk%7D%3D%20%5Cfrac%7B2%20%5Cpi%7D%7B0.71%20cm%5E%7B-1%7D%7D%3D8.8%20cm%20%20)
c) Period:
we can find the period by using the angular frequency:
![\omega= \frac{2 \pi}{T}= 36 s^{-1}](https://tex.z-dn.net/?f=%5Comega%3D%20%5Cfrac%7B2%20%5Cpi%7D%7BT%7D%3D%2036%20s%5E%7B-1%7D)
By re-arranging this relationship, we find
![T= \frac{2 \pi}{\omega}= \frac{2 \pi}{36 s^{-1}}=0.17 s](https://tex.z-dn.net/?f=T%3D%20%5Cfrac%7B2%20%5Cpi%7D%7B%5Comega%7D%3D%20%5Cfrac%7B2%20%5Cpi%7D%7B36%20s%5E%7B-1%7D%7D%3D0.17%20s%20%20)
d) Speed of the wave:
The speed of a wave is given by
![v= \lambda f](https://tex.z-dn.net/?f=v%3D%20%5Clambda%20f)
where f is the frequency of the wave, which is the reciprocal of the period:
![f= \frac{1}{T}= \frac{1}{0.17 s}=5.9 s^{-1}](https://tex.z-dn.net/?f=f%3D%20%5Cfrac%7B1%7D%7BT%7D%3D%20%5Cfrac%7B1%7D%7B0.17%20s%7D%3D5.9%20s%5E%7B-1%7D%20%20)
And so the speed of the wave is
![v= \lambda f=(8.8 cm)(5.9 s^{-1})=52 cm/s](https://tex.z-dn.net/?f=v%3D%20%5Clambda%20f%3D%288.8%20cm%29%285.9%20s%5E%7B-1%7D%29%3D52%20cm%2Fs)
e) Direction of the wave:
A wave written in the cosine form as
![y(x,t)=A \cos(\omega t- kx)](https://tex.z-dn.net/?f=y%28x%2Ct%29%3DA%20%5Ccos%28%5Comega%20t-%20kx%29)
propagates in the positive x-direction, while a wave written in the form
![y(x,t)=A \cos(\omega t+ kx)](https://tex.z-dn.net/?f=y%28x%2Ct%29%3DA%20%5Ccos%28%5Comega%20t%2B%20kx%29)
propagates in the negative x-direction. By looking at (2), we see we are in the second case, so our wave propagates in the negative x-direction.