1. Lecture 4: Longitudinal Waves
PHysics of music
Lecture 4: Longitudinal Waves
© Chris Waltham 2018

2. Informal Quiz
1. The "frequency" of a wave means: (a) How often a sound happens (b) The distance between two successive wave crests. (c) The time between two successive wave crests. (d) The number of wave crests that pass in a given time period. (e) How fast the wave crests go. (f) The time between a wave crest and a trough.
© Chris Waltham 2018

3. Informal Quiz
2. The “wavelength" of a wave: (a) is its position (b) is measured in Hz (c) is also called the amplitude (d) is the distance between two successive wave crests. (e) is the distance between two successive wave troughs. (f) is the number of wave crests that pass in a given time period.
© Chris Waltham 2018

4. Informal Quiz
3. In a transverse wave: (a) The particles of the medium travel at the wave speed. (b) The particles of the medium oscillate in the wave direction. (c) The particles of the medium oscillate at 90 degrees to the wave direction. (d) The particles of the medium migrate in the direction of the wave. (e) The particles of the medium migrate at 90 degrees to the wave direction. (f) The particle motion has nothing to do with the wave motion.
© Chris Waltham 2018

5. Longitudinal waves Same principles as transverse waves
Disturbance is in same direction as that of the wave
Conceptually harder to visualize
Instead of plotting y-position against x-position, we need to plot displacement in x-direction against x-position (confusing)
© Chris Waltham 2018

6. Acoustic Waves
Pressure disturbances in a fluid or solid propagate outward from the source as acoustic waves.
There are also accompanying density variations but we usually talk about pressure fluctuations as these can be measured more directly (with a microphone). For the simulations it is easy to see the density variations.
For a gas like air, pressure is proportional to density.
For sound in air, the pressure variations are generally tiny compared to the static absolute air pressure (i.e. that which is reported by weather stations).
Acoustic pressure refers to the relatively tiny fluctuations in the much larger static air pressure. An amplitude of a billionth of the static pressure can still be heard at audio frequencies.
© Chris Waltham 2018

7. gases At the macroscopic level, gases have Pressure Density
Temperature
When a gas is squeezed (think of a bicycle pump)
Pressure rises
You apply a force, and gas reacts with an opposite force (Newton III)
Density rises
You squeeze the same mass into a smaller volume, so the density goes up
Temperature rises
You do mechanical work on the system, so it heats up
© Chris Waltham 2018

8. Kinetic theory
The way gases behave can be understood entirely in terms of their physical composition.
Large numbers of very small particles (i.e. molecules, atoms) moving in random directions. This is thermal motion.
Colliding with each other and the walls of their container.
Air molecules (e.g. two atoms of nitrogen bound together) at room temperature have a mean speed of 500 m/s and travel an average of 0.1 μm before collision with another.
Atoms of a gas hugely magnified and slowed down. Some coloured red to aid tracking by eye.
Greg L at the English language Wikipedia [GFDL ( or CC-BY-SA-3.0 ( via Wikimedia Commons
© Chris Waltham 2018

9. Microscopic view of a sound wave
See Daniel Russell’s simulation:
The dots represent a tiny fraction of air molecules.
The thermal motion has been “frozen out” to avoid making a really confusing picture.
Look at the moving diaphragm on the left (in grey), imagine it colliding with an air molecule:
If it is moving toward the molecule, it will add kinetic energy to the recoiling molecule (like a baseball bat hitting a ball). These molecules will soon collide with others, and pass their extra energy on in a travelling wave.
The reverse is true: if it moving away from the molecule (its speed is much less than that of the molecule), the molecule will recoil with less kinetic energy than it had before. The molecules will deplete the energy of any others that they collide with (like a curling rock hitting a stationary rock).
© Chris Waltham 2018

10. Travelling Longitudinal wave simulation
Pick a dot and watch it for a while.
It is only going back and forth.
However, it is plain something is moving from left to right in the top plot.
...and right to left in the bottom plot.
© Chris Waltham 2018

11. Adding left and right waves
Pick a green dot and watch it for a while.
It is only going back and forth.
The whole wave pattern is also only going back and forth.
© Chris Waltham 2018

12. A complication: two kinds of nodes
The biggest density (pressure) changes occur where the displacements are minimum
The extreme case is when the wave hits a solid wall: the density varies maximally, but the particles cannot move.
So, density (pressure) antinodes are displacement nodes.
Likewise, density (pressure) nodes are displacement antinodes.
© Chris Waltham 2018

13. A complication: two kinds of nodes
Pressure (density) antinodes: A A A A A
nodes N N N N
Must always specify pressure/displacement when talking about (anti)nodes of longitudinal waves
Displacement (velocity) nodes: N N N N N
antinodes: A A A A
© Chris Waltham 2018

14. Standing waves in pipes
How many standing waves can be generated in a pipe?
In principle an infinite series of ever-increasing frequency and ever-decreasing wavelength
Which wavelengths will fit depends on whether the ends are open or closed
We will return to this more than once in the course of the term
© Chris Waltham 2018