Two waves traveling on a string in the same direction both have a frequency of 92 Hz, a wavelength of 0.19 m, and an amplitude of 0.32 m. What is the amplitude of the resultant wave if the original waves differ in phase by pi /3 rad?

Question

Two waves traveling on a string in the same direction both have a frequency of 92 Hz, a wavelength of 0.19 m, and an amplitude of 0.32 m. What is the amplitude of the resultant wave if the original waves differ in phase by pi /3 rad?

What is the phase difference between the two waves if the amplitude of the resultant wave is 0.25 m?

Answer 1

consider your trig

A1 *cos(wt+theta1) + A2 cos(wt+theta2)
= A3 * cos(wt+Theta3)
where
A3^2=(A1cosTheta1 +A2cosTheta2)^2+(A1sinTheta1+A2sinTheta2)^2

and Theta3 is given by

arc tan ( (A1SinTheta1+A2SinTheta2)/(A1cosTheta1+A2CosTheta1) )

Answer 2

To determine the amplitude of the resultant wave, we can use the principle of superposition. When two waves interfere constructively, their amplitudes add up. When they interfere destructively, their amplitudes cancel out.

To find the amplitude of the resultant wave, we can use the formula for the amplitude of interference between two waves:

A_resultant = √(A1^2 + A2^2 + 2 * A1 * A2 * cos(Δφ))

Where:
A_resultant is the amplitude of the resultant wave.
A1 and A2 are the amplitudes of the original waves.
Δφ is the phase difference between the two waves.

In this case:
A1 = A2 = 0.32 m (given)
Δφ = π/3 rad (given)

Substituting these values into the formula, we get:

A_resultant = √(0.32^2 + 0.32^2 + 2 * 0.32 * 0.32 * cos(π/3))

To evaluate the cosine value, we can recall that cos(π/3) = 0.5. Substituting this value, we have:

A_resultant = √(0.1024 + 0.1024 + 2 * 0.32 * 0.32 * 0.5)

A_resultant = √(0.2048 + 0.2048 + 0.1024)

A_resultant = √0.5112

A_resultant ≈ 0.715 m

So, the amplitude of the resultant wave is approximately 0.715 m.

Now, let's determine the phase difference between the two waves if the amplitude of the resultant wave is 0.25 m. To do this, we rearrange the formula:

cos(Δφ) = (A_resultant^2 - A1^2 - A2^2) / (2 * A1 * A2)

Substituting the given values:

A_resultant = 0.25 m (given)
A1 = A2 = 0.32 m (given)

cos(Δφ) = (0.25^2 - 0.32^2 - 0.32^2) / (2 * 0.32 * 0.32)

cos(Δφ) = (0.0625 - 0.1024 - 0.1024) / (0.2048)

cos(Δφ) = -0.1423

To find the phase difference Δφ, we can use the inverse cosine function (arccos):

Δφ = arccos(-0.1423)

Using a calculator, we find:

Δφ ≈ 1.717 radians

So, the phase difference between the two waves is approximately 1.717 radians.