Two sinusoidal waves travel in the same direction and have the same frequency. Their amplitudes are Y1 and Y2. The smallest possible amplitude of the resultant wave is:
a) Y1 + Y2 and occurs if they are 180 degrees out of phase
b)[Y1 - Y2] and occurs if the are 180 degrees out of phase
c) Y1 + Y2 and occurs if they are in phase
d) [y1-Y2] and occurs if they are in phase
e) [Y1-Y2] and occurs if they are 90 out of phase
My thoughts:
All the explainations of resultants in my book for waves show Y1+Y2=Resultant for the theory of superposition. So I think then to get the smallest wave it would be A. Y1 + Y2 if they are 180 degrees out of phase.
The displacement of a string is given by
y(x,t)=Ysin(Kx + wt)
speed of the wave is
a)2pik/w
b)w/K
c)wk
d)2pi/k
e)k/2pi
Work:
v=lamba*f
k=2pi/lambda
w=2pi/T
v=2pi/lambda * 2pi/(1/f)
v=w/k
so I choose b
Help is appreciated :) Just to explain on how to go about doing it. That way I can do the work and post if for corrections.
For the question about the smallest possible amplitude of the resultant wave, you are correct in thinking that the theory of superposition applies here. When two sinusoidal waves are 180 degrees out of phase (or have a phase difference of pi radians), they interfere destructively and cancel each other out, resulting in the smallest possible amplitude of the resultant wave. Therefore, option a) Y1 + Y2 is the correct answer.
Now let's move on to the question about the speed of the wave. The displacement of a string is given by the equation y(x, t) = Y sin(Kx + wt), where Y is the amplitude, K is the wave number (related to the wavelength), x is the spatial position, t is the time, and w is the angular frequency (related to the period, T).
Velocity (v) is defined as the rate of change of displacement with respect to time. In this case, velocity is equal to dw/dk, where w is the angular frequency and k is the wave number. To find the velocity, we need to differentiate y(x, t) with respect to x and t.
Differentiating with respect to x:
∂y/∂x = Y cos(Kx + wt)
Differentiating with respect to t:
∂y/∂t = Y w cos(Kx + wt)
To find the velocity, we divide the time derivative (∂y/∂t) by the spatial derivative (∂y/∂x):
v = (∂y/∂t) / (∂y/∂x)
= (Y w cos(Kx + wt)) / (Y cos(Kx + wt))
= w / K
Therefore, the velocity of the wave is equal to w/K. Since we are asked for the speed, which is defined as the magnitude of velocity, the correct answer is b) w/K.
I hope this explanation helps you understand the concepts and steps involved in solving these questions. Let me know if you have any further questions or need clarification on anything.