Question

Three sinusoidal waves travel in the positive \(x\) direction along the same string. All three waves have the same frequency. Their amplitudes are in the ratio \(1: \frac{1}{2}: \frac{1}{3}\) and their phase angles are \(0, \pi / 2\), and \(\pi\), respectively. Plot the resultant waveform and discuss its behavior as \(t\) increases.

Short answer

The resultant wave \(y\) is a sinusoidal wave that is a superposition of the three original waves with varying amplitudes and phase shifts. Its behavior over time will depend on these properties, resulting in changes in amplitude.

Step-by-step solution

Define the Equations for the Waves

Establish the mathematical representation of each wave. This can be done with the standard equation for a sinusoidal wave, which is \(y = A \sin(\omega t + \phi)\), where \(A\) is the amplitude, \(\omega\) is the angular frequency, \(t\) is the time, and \(\phi\) is the phase angle. For the three waves, the equations end up as: Wave 1: \(y_1 = \sin(\omega t)\), Wave 2: \(y_2 = 0.5\sin(\omega t + \frac{\pi}{2})\), and Wave 3: \(y_3 = \frac{1}{3}\sin(\omega t + \pi)\).

Superpose the Waves

Calculate the superposition of the waves, which will be the sum of the individual wave equations from step 1. The resultant wave \(y\) can be expressed as: \(y = y_1 + y_2 + y_3 = \sin(\omega t) + 0.5\sin(\omega t + \frac{\pi}{2}) + \frac{1}{3}\sin(\omega t + \pi)\).

Plot the Resultant Waveform

Plot the resulting waveform \(y\) as a function of time \(t\), using the equation from step 2. The graph should show changes in amplitude and frequency due to the superposition of the original sinusoidal waves.

Discuss the Behavior of the Resultant Waveform

Discuss the behavior of the resulting waveform as time \(t\) increases, based on the graph. The waveform should show sinusoidal behavior due to the nature of the original waves, but changes in amplitude will occur due to the superposition.

Key concepts

Sinusoidal Wave Equation

The sinusoidal wave equation models the shape and motion of waves, such as those moving along a string or sound waves travelling through the air. The basic form of this equation for a wave moving in the positive direction is given by
\[ y = A \sin(\omega t + \phi) \]
where \(y\) is the displacement at time \(t\), \(A\) represents the amplitude of the wave, \(\omega\) is the angular frequency, which determines how many oscillations occur per unit time, and \(\phi\) is the phase angle, affecting the starting position of the wave. When multiple waves with the same frequency but varying amplitudes and phases travel together, as in the exercise, their individual contributions are added together to form a complex waveform—this process is known as superposition.

Phase Angle

The phase angle, denoted by \(\phi\), dictates where in its cycle a wave begins at \(t = 0\). If the phase angle is zero, the sinusoidal wave starts at the origin and moves in the positive direction. A phase angle of \(\pi/2\) means the wave starts at its peak amplitude, while a phase angle of \(\pi\) corresponds to the wave starting at the origin and moving in the negative direction.
These shifts are crucial when combining waves, as they determine how the peaks and troughs of the different waves align, affecting the overall shape and properties of the resultant waveform. In the given exercise, varying phase angles lead to different starting points, which in turn influences the pattern observed when the waves are combined.

Angular Frequency

Angular frequency, usually represented by \(\omega\), is a measure of how quickly the wave oscillates. It is related to the ordinary frequency \(f\) (the number of cycles per second) by the equation \[ \omega = 2\pi f \].
In our exercise, all waves have the same angular frequency; this means they have the same rate of oscillation, making the wave patterns more straightforward to analyze as time passes. Every point on the wave undergoes the same number of cycles per unit time. The angular frequency's consistency across waves ensures the superposition results in a periodically repeating pattern.

Wave Interference

Wave interference occurs when two or more waves superpose to form a resulting wave of greater, lower, or the same amplitude. Interference can be constructive, destructive, or a complex mix, depending on their phase angles and amplitudes.

In constructive interference, waves combine to produce a wave with a greater amplitude, while in destructive interference, they produce a wave with a lower amplitude or even cancel each other out. The exercise demonstrates a complex interference pattern due to the superposition of three waves with different amplitudes and phase angles. As time increases, the combined effect of these factors leads to a more complicated resultant waveform that still maintains a sinusoidal nature but with varying amplitude.