Two waves traveling on a string in the same direction both have a frequency of 100 Hz, a wavelength of 2cm and an amplitude of .02 m. What is the amplitude of the resultant wave if the original waves differ in phase (a) by Pi/6 and (b) by Pi/3

To find the amplitude of the resultant wave when two waves with different phases are superimposed, we can use the concept of interference.

Interference occurs when waves combine to form a new wave. In this case, when two waves are traveling on a string in the same direction, their amplitudes can either add up constructively (resulting in a higher amplitude) or interfere destructively (resulting in a lower amplitude). The phase difference between the two waves determines the type of interference.

(a) When the phase difference between the waves is π/6:

Constructive interference occurs when the phase difference is an even multiple of π (180 degrees). Since π/6 is less than π (180 degrees), constructive interference will happen.

The formula to calculate the amplitude of the resultant wave for constructive interference is:

Amplitude_resultant = Amplitude_wave1 + Amplitude_wave2

Given that the amplitude of both waves is 0.02 m, the amplitude of the resultant wave is:

Amplitude_resultant = 0.02 m + 0.02 m = 0.04 m

Therefore, the amplitude of the resultant wave when the phase difference is π/6 is 0.04 m.

(b) When the phase difference between the waves is π/3:

Destructive interference occurs when the phase difference is an odd multiple of π/2 (90 degrees). Since π/3 is not a multiple of π/2, destructive interference will not occur.

To find the amplitude of the resultant wave when there is no destructive interference, we need to calculate the resultant amplitude using the formula:

Amplitude_resultant = √((Amplitude_wave1)² + (Amplitude_wave2)² + 2(Amplitude_wave1)(Amplitude_wave2)cos(δ))

Where δ is the phase difference between the waves.

Given that the amplitude of both waves is 0.02 m, the phase difference δ is π/3, and substituting the values into the formula, we can calculate:

Amplitude_resultant = √((0.02 m)² + (0.02 m)² + 2(0.02 m)(0.02 m)cos(π/3))
Amplitude_resultant = √(0.0004 m² + 0.0004 m² + 2(0.02 m)(0.02 m)(0.5))
Amplitude_resultant = √(0.0008 m² + 0.0008 m² + 0.0008 m²)
Amplitude_resultant = √(0.0024 m²)
Amplitude_resultant ≈ 0.049 m

Therefore, the amplitude of the resultant wave when the phase difference is π/3 is approximately 0.049 m.

To find the amplitude of the resultant wave, we need to consider the superposition of the two waves. The superposition principle states that when two waves meet, their displacements add up to form a resultant wave.

Given:
Frequency of both waves: 100 Hz
Wavelength of both waves: 2 cm
Amplitude of both waves: 0.02 m

Let's calculate the amplitude of the resultant wave for both cases:

(a) The waves differ in phase by π/6:
When the phase difference is π/6, we can consider the waves as signals with a phase shift. In this case, the resultant amplitude can be calculated using trigonometric functions. The formula for the resultant amplitude, A(r), can be given as:

A(r) = 2 * √(A^2 * (1 + cosθ))

Where:
A = Amplitude of the individual waves
θ = Phase difference between the waves

Given: A = 0.02 m and θ = π/6

Plug in the values:

A(r) = 2 * √(0.02^2 * (1 + cos(π/6)))

Calculating the cosine of π/6:

A(r) = 2 * √(0.02^2 * (1 + √3/2))

A(r) = 2 * √(0.0004 * (1 + √3/2))

A(r) = 2 * √(0.0004 * (2 + √3))

A(r) = 2 * √(0.0004 * 2 + 0.0004 * √3)

A(r) = 2 * √(0.0008 + 0.0004√3)

A(r) = 2 * √(0.0008 + 0.001732)

A(r) = 2 * √(0.002532)

A(r) ≈ 0.0893 m

The amplitude of the resultant wave when the phase difference is π/6 is approximately 0.0893 meters.

(b) The waves differ in phase by π/3:
Using the same formula and calculations as above, with θ = π/3, we can find the amplitude of the resultant wave.

A(r) = 2 * √(A^2 * (1 + cosθ))

Given: A = 0.02 m and θ = π/3

A(r) = 2 * √(0.02^2 * (1 + cos(π/3)))

Calculating the cosine of π/3:

A(r) = 2 * √(0.02^2 * (1 + 1/2))

A(r) = 2 * √(0.0004 * (1 + 1/2))

A(r) = 2 * √(0.0004 * (3/2))

A(r) = 2 * √(0.0006)

A(r) ≈ 0.0346 m

The amplitude of the resultant wave when the phase difference is π/3 is approximately 0.0346 meters.