Waves

Waves
A Brief Introduction

Computer Science Department

bernstdh@jmu.edu

Mechanical Waves

Pulse:
- A single disturbance that moves through a sequence of interacting particles (called a medium)
Periodic Wave:
- A periodic disturbance that moves through a medium, transporting energy as it moves

Mechanical Waves (cont.)

Some Observations:
- Individual particles do not move very far (each particle oscillates around its equilibrium position, but its average position does not change)
- As a particle interacts with its neighbors it transfers some of its energy to them, causing them to oscillate
- As this process continues, the energy is transported throug the medium.
Common Examples:
- Waves in water (from which you have developed most of your intuition)
- Waves in a spring (a medium consisting of individual coils)

Mechanical Waves (cont.)

One way to generate a wave:
- Move the left end of the spring "back and forth" in the horizontal direction
The result - a longitudinal wave:
- A series of compressions (areas in which the particles are closer than in equilibrium) and rarefactions (areas in which the particles are farther apart than in equilibrium)

Mechanical Waves (cont.)

Another way to generate a wave:
- Move the left end of the spring "up and down" in the vertical direction
The result - a transverse wave:
- A series of peaks (areas in which the particles are "higher" than in equilibrium) and troughs (areas in which the particles are "lower" than in equilibrium)

The Position Domain

The Position Domain (cont.)

The Position Domain (cont.)

Wavelength (of a periodic wave):
- The distance one has to travel along the wave until it "repeats"
- Usually denoted by \(\lambda\) (and measured in some unit of length per cycle)

The Position Domain (cont.)

Interference:
- When two waves meet while traveling through the same medium they are said to interfere with each other
Principle of Superposition:
- When two waves interfere, the resulting displacement of the medium at any location is the sum of the displacements of the individual waves at that same location
Types of Interference:
- Constructive
- Destructive

The Time Domain

The Concept:
- The figures above are all plots of the amplitude of the wave versus the position
- Alternatively we could have picked a particular position along the wave and looked at how the amplitude at that position changed over time
An Example:

The Time Domain (cont.)

Cycle:
- A portion of a wave from that goes from rest to crest to rest to trough to rest
Period:
- The time required for a cycle
Frequency:
- The reciprocal of the period (i.e., is the number of cycles per second or hertz)
- Usually denoted by \(f\)
Speed:
- The product of wavelength and frequency
- Usually denoted by \(v\)
- Thinking about units: \(v = \lambda f\) (e.g., m/sec = m/cycle \(\cdot\) cycles/second)

The Frequency Domain

A 400Hz Periodic Wave:
Spectra:
- Periodic waves have a line spectrum
- Quasiperiodic waves have harmonic spectra
- Aperiodic waves have continuous spectra

The Frequency Domain (cont.)

The Fourier Transform:
- Can be used to convert from the time domain to the frequency domain
Fourier's Discovery:
- All periodic waves may be expressed as the sum of a series of sinusoidal waves
- These waves are all integer multiples (called harmonics) of the fundamental frequency
- Each harmonic has its own amplitude and phase

The Doppler Effect

Defined:
- An increase (or decrease) in the frequency of a wave as the source and observer move toward (or away from) each other
History:
- Described by Christian Doppler in 1842

A Sound Wave Example of the Doppler Effect

Situation 1:
- The Source: A car 1 mile away, standing still, plays a pure tone for one minute
- The Observer: Hears nothing for 4.73 secs (while the wave travels the one mile at 761 mph) and then hears the tone for one minute
Situation 2:
- The Source: A car 1 mile away, traveling towards the observer at 60mph, plays a pure tone for one minute
- The Observer: Hears nothing for 4.73 secs (while the wave travels the one mile at 761 mph) and then hears something for 55.27 seconds

A Sound Wave Example of the Doppler Effect (cont.)

Understanding Situation 2:
- It takes the car 1 minute (i.e., 60 seconds) to travel the mile
- So, the tone stops being played exactly when the car reaches the observer
- So, the tone stops being played 55.27 seconds after it is first heard by the observer
- Since the car is located at the observer when the tone stops playing, the observer stops hearing it exactly when it stops being played
Important Observations:
- The tone is being played for 60 seconds but only heard by the observer for 55.27 seconds
- This means that the same number of waves reach the observer in a smaller amount of time, which is to say that the frequency of the waves must have increased

The Doppler Effect (cont.)

Notation:
- \(f_o\) denotes the observed frequency
- \(f_e\) denotes the emitted frequency
- \(v_w\) denotes the velocity of the wave in the medium
- \(v_s\) denotes the velocity of the source
One Case:
- The source is moving towards the observer
- The observer is stationary
The Relationship:
- \(f_o = \frac{v_w}{v_w - v_s} f_e\)

The Doppler Effect (cont.)

The Doppler Effect (cont.)

An Illustration

Source: Wikimedia

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