Two sinusoidal waves of the same frequency are to be sent in...

Two sinusoidal waves of the same frequency are to be sent in the same direction along a taut string. One wave has an amplitude of $5.0 \mathrm{~mm}$, the other $8.0 \mathrm{~mm}$. (a) What phase difference $\phi_{1}$ between the two waves results in the smallest amplitude of the resultant wave? (b) What is that smallest amplitude? (c) What phase difference $\phi_{2}$ results in the largest amplitude of the resultant wave? (d) What is that largest amplitude? (e) What is the resultant amplitude if the phase angle is $\left(\phi_{1}-\phi_{2}\right) / 2 ?$

Key Concepts

Superposition Principle

This principle states that when two or more waves overlap in space, the total displacement at any point is the sum of the individual displacements. It is fundamental in understanding how waves interact, leading to resulting patterns known as interference, whether constructive or destructive.

Phase Difference

Phase difference is the measure of the shift in the phase of one wave relative to another, usually expressed in radians. It is a crucial factor in wave interference as it determines whether the waves reinforce each other (constructive interference) or cancel each other out (destructive interference).

Constructive Interference

Constructive interference occurs when two waves are in phase or have a phase difference of 0 or an integer multiple of 2?. Under such conditions, the amplitudes of the individual waves add up, resulting in a larger resultant amplitude than the individual waves.

Destructive Interference

Destructive interference takes place when the waves are out of phase by ? (or an odd multiple of ?). In this scenario, the waves subtract from each other, leading to a decreased resultant amplitude and, in perfect cancellation, to a net amplitude of zero.

Resultant Amplitude Calculation

The resultant amplitude when combining two sinusoidal waves of the same frequency can be determined through vector (phasor) addition, accounting for both the individual amplitudes and their phase difference. The general formula is A_resultant = ?(A1² + A2² + 2A1A2 cos(?)), where A1 and A2 are the amplitudes of the individual waves and ? is their phase difference.

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