Question

If two waves with the same frequency are \(180^{\circ}\) out of phase, what is the amplitude of the resultant wave if the amplitudes of the original waves are \(5 \mathrm{cm}\) and \(3 \mathrm{cm} ?\) (A) \(2 \mathrm{cm}\) (B) \(3 \mathrm{cm}\) (C) \(\quad 5 \mathrm{cm}\) (D) \(8 \mathrm{cm}\)

Short answer

2 cm

Step-by-step solution

- Understanding Out of Phase

Two waves are said to be 180 degrees out of phase if one wave is the complete inverse of the other. This means that when one wave reaches its maximum positive value, the other wave reaches its maximum negative value.

- Calculate Amplitude of Resultant Wave

Since the waves are 180 degrees out of phase, their amplitudes can be subtracted to find the amplitude of the resultant wave. The formula for the resultant amplitude (\text{A}_{\text{resultant}}) when two waves are out of phase is given by: \[ \text{A}_{\text{resultant}} = |A_1 - A_2| \] where \( A_1 \) and \( A_2 \) are the amplitudes of the two waves. Here, \( A_1 = 5 \text{ cm} \) and \( A_2 = 3 \text{ cm} \).

- Substitute Amplitudes into Formula

Substitute the given amplitudes into the formula: \[ \text{A}_{\text{resultant}} = |5 \text{ cm} - 3 \text{ cm}| \] which simplifies to: \[ \text{A}_{\text{resultant}} = |2 \text{ cm}| = 2 \text{ cm} \]

- Verify the Answer

Verify the calculation to ensure it's correct. Based on the steps above, the amplitude of the resultant wave is 2 cm.

Key concepts

Phase Difference

Phase difference is a crucial concept in understanding wave interference. When we talk about phase difference, we're referring to how much one wave is shifted from another along the horizontal axis. For instance, if two waves have a phase difference of 180 degrees, they are essentially shifted by half a wavelength relative to each other. This means that when one wave reaches its peak, the other wave reaches its trough. In such cases, the waves are described as being 'out of phase'. The complete inversion causes significant effects, such as in the problem given, where two waves with amplitudes of 5 cm and 3 cm are 180 degrees out of phase.

Resultant Amplitude

The resultant amplitude is the combined effect of two or more waves interfering with each other. When dealing with waves that are out of phase, like the ones in this problem, the resultant amplitude is found by calculating the difference in their individual amplitudes. Mathematically, for waves out of phase by 180 degrees, the resultant amplitude \(A_{\text{resultant}} \) is given by: \[ \ A_{\text{resultant}} = |A_1 - A_2| \ \] where \(A_1 = 5 \) cm and \(A_2 = 3 \) cm. When we plug in these values, we get: \[ \ \ A_{\text{resultant}} = |5 \text{ cm} - 3 \text{ cm}| \ \ \ = 2 \text{ cm} \ \] Thus, the resultant wave has an amplitude of 2 cm.

Wave Superposition

Wave superposition refers to the principle that when two or more waves overlap, their effects add together. This can result in constructive interference, where the waves reinforce each other, or destructive interference, where the waves cancel each other out. In the exercise, the waves are 180 degrees out of phase, leading to destructive interference. Destructive interference occurs when the positive displacement of one wave aligns with the negative displacement of another, reducing the overall amplitude. This is why we subtract the amplitudes rather than add them, leading to a smaller resultant amplitude.

Amplitude Subtraction

Amplitude subtraction occurs when we calculate the resultant amplitude of waves that are out of phase. In the example problem, the waves with amplitudes of 5 cm and 3 cm are 180 degrees out of phase. This phase relationship means that we subtract the smaller amplitude from the larger one: \[ \ A_{\text{resultant}} = |A_1 - A_2| \] Substituting the given values yields: \[ \ A_{\text{resultant}} = |5 \text{ cm} - 3 \text{ cm}| = 2 \text{ cm} \ \] This resultant amplitude of 2 cm is the answer. Understanding amplitude subtraction helps in predicting the outcomes of wave interferences in various practical and theoretical scenarios.