Two sinussoidal waves identical except for phase travel in the same direction along a string and interfere to produce a resultant wave given by y(x,t)= (3.0mm) sin(20 rad/m)x - 4.0rad/s t + 0.820 rad) with x in meters and t in seconds. What is a) the wavelength of the two waves b) the phase difference between them c) their amplitude
Work:
I know the resultant of the waves comes from the formula
y(x,t)=y(x,t) + y2(x,t)
since there out of phase
y(x,t)= {2Ycos1/2phase}sin(kx-wt + 1/2phase)
I tried to figure out the equations but I am lost. I read the book and its confusing more on this section.Help is appreciated. Thank you :)
Well, you almost have it.
y(x,t)= {2Ycos1/2phase}sin(kx-wt+1/2phase)
3mm= 2Ycos1/2phase
and 1/2 phase= .820radians.
Solve for phase, then Y.
you know k= 20, and w=4
solve for w, and if you wish, k. k ought to related to wavelength. Finally, knowing w, you have f.
To solve this problem, we can start by analyzing the given equation for the resultant wave:
y(x, t) = (3.0 mm) sin(20 rad/m)x - (4.0 rad/s)t + 0.820 rad)
a) The wavelength of the waves:
In the equation y(x, t) = (2Y cos(1/2 phase)) sin(kx - wt + 1/2 phase), we can see that the wave number, k, is equal to 20 rad/m. The wave number is related to the wavelength λ by the formula λ = 2π/k.
Thus, we can calculate the wavelength of the waves by substituting the given value of k into the formula:
λ = 2π/(20 rad/m)
λ = π/10 rad/m
So, the wavelength of the two waves is π/10 rad/m.
b) The phase difference between the waves:
From the given equation, we know that 1/2 phase = 0.820 rad. We can solve for the phase as follows:
1/2 phase = 0.820 rad
phase = 2 * 0.820 rad
phase = 1.640 rad
Therefore, the phase difference between the waves is 1.640 rad.
c) The amplitude of the waves:
From the equation, we have the relation 3.0 mm = 2Y cos(1/2 phase). We can solve for Y by rearranging the equation:
2Y cos(1/2 phase) = 3.0 mm
Y = (3.0 mm) / (2 cos(1/2 phase))
Substituting the value of 1/2 phase, we can calculate the amplitude:
Y = (3.0 mm) / (2 cos(0.820 rad))
Y ≈ 1.739 mm
Therefore, the amplitude of the waves is approximately 1.739 mm.
To find the angular frequency, ω, we can relate it to the given angular speed, w, by the formula ω = 2πf, where f is the frequency. Since w = 4 rad/s, we have:
ω = 2π(4 rad/s)
ω = 8π rad/s
So, the angular frequency of the waves is 8π rad/s.