AdvancedTopic 74 of 104Engineering Mathematics
Wave equation
u_tt = c² u_xx models vibrating strings, sound and electromagnetic waves.
In depth
u_tt=c²u_xx has solutions u=f(x-ct)+g(x+ct): waves travelling both ways at speed c. A string fixed at both ends has standing waves sin(nπx/L)cos(nπct/L) with frequencies n·c/(2L). Applies to sound, strings and electromagnetic waves.
Key points
- d'Alembert solution
- Harmonics are integer multiples
- Dispersion changes speed with frequency
ExampleA guitar string of length 0.65 m and c=300 m/s has fundamental near 230 Hz.
The wave equation describes vibrating strings, sound and electromagnetic waves, so it underpins acoustic and radio design.
What is Wave equation?
u_tt = c² u_xx models vibrating strings, sound and electromagnetic waves.
Why is it important to understand wave equation?
The wave equation describes vibrating strings, sound and electromagnetic waves, so it underpins acoustic and radio design.
What sets the wave speed in the equation?
The constant c is the speed of propagation, given by the medium's properties, such as tension over linear density for a string.
What is a standing wave?
A pattern formed by two waves travelling in opposite directions, with fixed nodes and antinodes. It occurs at the natural frequencies of a bounded system.
How is Wave equation used in practice?
Typical applications include calculating the speed of a pulse along a rope or cable, predicting the resonant frequencies of a guitar string or pipe and designing the spacing of antennas in a radio array.
What should I learn before Wave equation?
Study "Heat equation" first (topic 73 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Wave equation?
d'Alembert solution. Harmonics are integer multiples. Dispersion changes speed with frequency.
Can you give an example of Wave equation?
A guitar string of length 0.65 m and c=300 m/s has fundamental near 230 Hz.
What comes after Wave equation?
The next topic is "Separation of variables for PDEs", which builds directly on this one.
How difficult is Wave equation?
It is classed as Advanced in Engineering Mathematics (topic 74 of 104). It is an advanced topic. It assumes you are comfortable with the intermediate topics in Engineering Mathematics.