Suppose two linear waves of equal amplitude and frequency have a phase difference ϕ as they travel in the same medium. They can be represented by: D₁ = A sin (kx - ωt); D₂ = A sin ( kx - ωt + ϕ). What is the amplitude of this resultant wave? Is the wave purely sinusoidal, or not?

Verified step by step guidance

1. Step 1: Represent the two waves mathematically. The first wave is given as \( D_1 = A \sin(kx - \omega t) \), and the second wave is \( D_2 = A \sin(kx - \omega t + \phi) \), where \( A \) is the amplitude, \( k \) is the wave number, \( \omega \) is the angular frequency, and \( \phi \) is the phase difference.
2. Step 2: Use the principle of superposition to find the resultant wave. The total displacement \( D \) is the sum of the two waves: \( D = D_1 + D_2 = A \sin(kx - \omega t) + A \sin(kx - \omega t + \phi) \).
3. Step 3: Apply the trigonometric identity for the sum of two sine functions: \( \sin(x) + \sin(y) = 2 \sin\left(\frac{x + y}{2}\right) \cos\left(\frac{x - y}{2}\right) \). Substituting \( x = kx - \omega t \) and \( y = kx - \omega t + \phi \), we get \( D = 2A \cos\left(\frac{\phi}{2}\right) \sin\left(kx - \omega t + \frac{\phi}{2}\right) \).
4. Step 4: Identify the amplitude of the resultant wave. From the expression above, the amplitude of the resultant wave is \( 2A \cos\left(\frac{\phi}{2}\right) \). This shows that the amplitude depends on the phase difference \( \phi \).
5. Step 5: Determine if the wave is purely sinusoidal. The resultant wave is of the form \( D = \text{(amplitude)} \times \sin(\text{(argument)}) \), which is a sinusoidal function. Therefore, the wave is purely sinusoidal.

Key Concepts

Superposition Principle

The superposition principle states that when two or more waves overlap in the same medium, the resultant displacement at any point is the sum of the displacements due to each wave. This principle is fundamental in wave theory and allows us to analyze complex waveforms by breaking them down into simpler components.

Phase Difference

Phase difference refers to the difference in phase angle between two waves at a given point in time. It affects how the waves interfere with each other, leading to constructive or destructive interference, which ultimately influences the amplitude and shape of the resultant wave.

Resultant Amplitude

The resultant amplitude of two interfering waves can be calculated using the formula that incorporates their individual amplitudes and the phase difference. For waves of equal amplitude, the resultant amplitude can be expressed as A_r = 2A cos(ϕ/2), where ϕ is the phase difference, indicating that the resultant wave's amplitude varies depending on the phase relationship between the two waves.