Question

Four sinusoidal waves travel in the positive \(x\) direction along the same string. Their frequencies are in the ratio \(1: 2: 3: 4\) and their amplitudes are in the ratio \(1: \frac{1}{2}: \frac{1}{3}: \frac{1}{4}\), respectively. When \(t=0\), at \(x=0\), the first and third waves are \(180^{\circ}\) out of phase with the second and fourth. Plot the resultant waveform when \(t=0\) and discuss its behavior as \(t\) increases.

Short answer

The composite waveform at \( t = 0 \) is constructed by combining the individual waves. The waveform's behavior as \( t \) increases is derived from the combined influence of the four sinusoidal waves, taking into account their amplitudes, frequencies, and phase differences. The result will be a complex waveform that reflects these combined influences.

Step-by-step solution

Define the Waves

Firstly, define the sinusoidal waves. The general form of a sinusoidal wave is given by \( y = A \sin \omega t \) where A is the amplitude, \(\omega\) is the angular frequency. As given the ratios of frequencies and amplitudes for four waves are \(1:2:3:4\) and \(1:\frac{1}{2}:\frac{1}{3}:\frac{1}{4}\) respectively. Let's assume the base frequency is \( f \) and base amplitude is \( a \). The waves will be \( y_1= a \sin 2\pi f t \) for the 1st wave, \( y_2=\frac{1}{2}a \sin 4\pi f t \) for the 2nd wave, \( y_3=\frac{1}{3}a \sin 6\pi f t \) for the 3rd wave and \( y_4=\frac{1}{4}a \sin 8\pi f t \) for the 4th wave.

Apply Phase Shift

Next, apply the phase shift to the first and third waves. Given that the first and third waves are \(180^\circ\) out of phase with the second and fourth waves, this phase shift is translated as \( \pi \) in radian measure. Hence the equations becomes \( y_1= a \sin (2\pi f t + \pi) \) , \( y_2=\frac{1}{2}a \sin 4\pi f t \), \( y_3=\frac{1}{3}a \sin (6\pi f t + \pi) \) and \( y_4=\frac{1}{4}a \sin 8\pi f t \).

Combine the Waves

Now, add all the waves together to get the resultant waveform. The resultant waveform at \( t = 0 \) will be \( y = y_1 + y_2 + y_3 + y_4 \). Substitute the value of \( t = 0 \) into this equation to get the resultant waveform at \( t = 0 \)

Analyze the Result

Discuss the behaviour of this waveform as \( t \) increases. Consider characteristics such as frequency, amplitude, phase etc. of the resultant waveform and how these properties change with time.

Key concepts

Sinusoidal Waves

Sinusoidal waves are a fundamental concept in understanding wave phenomena. They are waves that describe a sine or cosine function in terms of time or space. A sinusoidal wave can be expressed as \( y = A \sin(\omega t) \), where:

• \( y \) is the wave's displacement at any point in space and time.
• \( A \) is the wave amplitude, depicting the maximum displacement from rest.
• \( \omega \) is the angular frequency, indicating how quickly the wave oscillates in radians per unit time.

These waves are important in studying vibrations and oscillations, including sound waves and electromagnetic waves. Due to their repetitive nature, they exhibit periodic motion, meaning they repeat after a specific interval known as the wavelength. The sequence determines the wave's behavior and structure, making it predictable and regular.
Aside from basic properties, sinusoidal waves can be added together (superposed) to produce complex waveforms. This principle is essential in many scientific fields, from engineering to music theory.

Phase Shift

Phase shift is a concept that refers to the horizontal shift of a wave along the time axis. When two waves are out of phase, they are not aligned exactly at the same frequency or time. This is important because a shift of \(180^\circ\) or \(\pi\) in radians results in the wave being in complete out-of-phase status with another wave. This means one wave reaches its peak when the other reaches its trough, leading them to cancel each other out in terms of their effect.
In our exercise, the first and third waves have a phase shift of \( \pi \) radians relative to the second and fourth waves. This phase shift has a significant impact on the resultant waveform when all these waves are combined. The fundamental idea is that when such phase shifts occur, the resultant wave can significantly differ in shape and behavior compared to any of the individual waves.

Wave Amplitude

Amplitude in waves refers to the maximum extent of vibration or oscillation, measured from the position of equilibrium. In a sinusoidal wave, it determines how far the medium's particles are displaced from their central position. Higher amplitude indicates a stronger wave.
For the waves discussed in the given exercise, amplitudes were given in decreasing ratios. Specifically:

• The first wave has the highest amplitude \( a \).
• The second wave has amplitude \( \frac{1}{2}a \).
• The third wave has amplitude \( \frac{1}{3}a \).
• The fourth wave has amplitude \( \frac{1}{4}a \).

Amplitude is crucial because it affects the wave's energy. More amplitude means more energy, which can be seen in how waves interact when superimposed. Variations in amplitude lead to different resultant waveforms when combined.

Wave Frequency

Frequency refers to the number of cycles a wave completes in one second. It is measured in hertz (Hz) and is an indicator of how fast a wave oscillates. In the exercise, the frequencies of four waves are provided in the ratios \( 1: 2: 3: 4 \). This indicates that, relative to a base frequency \( f \):

• The first wave has frequency \( f \).
• The second wave has frequency \( 2f \).
• The third wave has frequency \( 3f \).
• The fourth wave has frequency \( 4f \).

Frequency directly impacts the wave's properties, such as wavelength and energy. A higher frequency involves more cycles per second, translating to shorter wavelengths. Understanding frequency is essential as it affects how the resultant waveform evolves and behaves over time, particularly when waves of different frequencies are combined.