How Do Phase Shifts Affect Resultant Wave Amplitude and Frequency?

[Husker70]

Homework Statement

Two traveling sinusoidal waves are described by the wave functions
y1 = (5.00m) sin[pie(4.00x - 1 200t)]
y2 = (5.00m) sin[pie(4.00x - 1 200t -0.250y)]
where x, y1, and y2 are in meters and t is in seconds.
(a) What s the amplitude of the resultant wave?
(b) What is the frequency of the resultant wave?

Homework Equations

y = (2Acos Phi/2) sin(kx- wt + Phi/2)

The Attempt at a Solution

The two waves are out of phase by 0.250
therefore A = (2A cos Phi/2)
I get the Amplitude to be 9.99 or 10m
from 2 (5.00m) .25/2
Do I need to do more for this? I know the answer but can't get there
Thanks,
Kevin

 

[Redbelly98]

Homework Statement

Two traveling sinusoidal waves are described by the wave functions
y1 = (5.00m) sin[pie(4.00x - 1 200t)]
y2 = (5.00m) sin[pie(4.00x - 1 200t -0.250y)]
where x, y1, and y2 are in meters and t is in seconds.

Is it "-0.250y" or just "-0.250" for the phase in y2?

(a) What s the amplitude of the resultant wave?
(b) What is the frequency of the resultant wave?

Homework Equations

y = (2Acos Phi/2) sin(kx- wt + Phi/2)

The Attempt at a Solution

The two waves are out of phase by 0.250
therefore A = (2A cos Phi/2)
I get the Amplitude to be 9.99 or 10m

It should be slightly lower. Was your calculator in radians mode when you calculated cos(0.25)?

from 2 (5.00m) .25/2
Do I need to do more for this? I know the answer but can't get there
Thanks,
Kevin

 

[thomate1]

Dear Kevin,

Thank you for your question. I am happy to help you with this problem.

First, let's take a closer look at the given equations for y1 and y2. Both equations have the same amplitude of 5.00m, which means that the maximum displacement of the waves is 5.00m. However, the second equation (y2) has an additional phase shift of -0.250y. This phase shift means that the wave is shifted by 0.250 wavelength in the negative y direction. This results in a different waveform compared to y1.

Now, to find the amplitude of the resultant wave, we need to use the principle of superposition. This principle states that when two waves meet, the resulting wave is the sum of the individual waves. In this case, the two waves have the same amplitude and frequency, but different phase shifts.

Using the equation you provided, y = (2Acos Φ/2) sin(kx- ωt + Φ/2), we can calculate the amplitude of the resultant wave. Since both waves have the same amplitude and frequency, we can write the equation as:

y = (2Acos Φ/2) sin(kx- ωt + Φ/2) + (2Acos Φ/2) sin(kx- ωt + Φ/2)

= 2(2Acos Φ/2) sin(kx- ωt + Φ/2)

= 4Acos Φ/2 sin(kx- ωt + Φ/2)

= 4(5.00m)cos 0.250y sin(pix - 1200t + 0.250y)

= 20cos 0.250y sin(pix - 1200t + 0.250y)

= 20cos 0.250y sin(pix - 1200t)cos 0.250y + 20sin 0.250y cos(pix - 1200t)

= 20cos 0.250y [sin(pix)cos(1200t) - cos(pix)sin(1200t)] + 20sin 0.250y [cos(pix)cos(1200t) + sin(pix)sin(1200t)]

= 20cos 0