Physics I · 120 h · Topic 7 of 10

Mechanical waves

A wave carries energy without carrying matter. A string with one fixed end and a short-circuited transmission line produce exactly the same standing wave.

Wave equation Standing waves Interference Sound Doppler

01What a wave is

A wave carries energy and momentum without carrying matter. Each particle of the medium oscillates about its equilibrium position, and what travels is the disturbance.

Transverse: the particles oscillate perpendicular to the direction of propagation. A string, an electromagnetic wave.
Longitudinal: they oscillate in the same direction in which the wave advances. Sound.

The fundamental relation

\[ v = \lambda \cdot f \]

The speed is set by the medium; the frequency, by the source. The wavelength is what results from the two. That is why, on passing from one medium to another, λ changes but f never does.

Medium	Speed	What it depends on
Stretched string	\( v = \sqrt{T/\mu } \)	Tension and mass per unit length
Air (sound, 20 °C)	343 m/s	Temperature, not pressure
Water	\( \approx 1500 \dfrac{m}{s} \)	Compressibility and density
Coaxial cable	\( \approx 0.66\cdot c \)	Permittivity of the dielectric

02The wave equation

A harmonic wave traveling to the right is written:

\[ y (x, t) = A \, \operatorname{sin} (k x -\omega t) , \quad k = \frac{2 \pi}{\lambda}, \quad \omega = 2 \pi f \] k is the wavenumber and ω the angular frequency. Their quotient, ω/k, is the propagation speed.

And all of them are solutions of one and the same partial differential equation:

\[ \frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2} \] The same equation that comes out of combining Maxwell’s equations, and the same one as for transmission lines.

That coincidence is no accident, and it has an enormous practical consequence: everything learned about a string holds for a cable. Reflection, impedance, standing waves and matching are the same phenomenon with different units.

03Superposition and standing waves

When two waves coincide at a point, the displacements add. If two equal waves travel in opposite directions —which happens naturally when one is reflected at an end— the result no longer advances: it is a standing wave, with points that never move, the nodes, and points of maximum oscillation, the antinodes.

\[ f_n = \frac{n v}{2 L}, \quad \lambda_n = \frac{2 L}{n} \] For a string with both ends fixed. Only those frequencies survive: they are the normal modes.

The lowest frequency, with n = 1, is the fundamental; the others are its harmonics, and the mix of their intensities is what gives each instrument its timbre.

Lab · modes of a string with fixed ends

Choose the harmonic and move through time. With the checkbox you can see the two traveling waves that, when added, produce the standing wave.

Harmonic n = 3

Length of the string 1.20 m

Propagation speed 120 m/s

Time, as a fraction of the period 0.00

Show the two traveling waves that add up

04Reflection and interference

End	What happens	Electrical equivalent
Fixed	The wave returns inverted	Short-circuited line
Free	It returns without inversion	Open-circuited line
Matched	Nothing is reflected: all the energy is absorbed	Load equal to the characteristic impedance

That third case is the one always sought in radio frequency, and the whole Smith chart exists to achieve it. The standing wave ratio of a poorly matched line is, exactly, the same pattern of nodes and antinodes as on the string.

Interference and beats

Two waves of the same frequency reinforce each other if they arrive in phase and cancel if they arrive in antiphase: hence the sound cancellations in a room and the nulls of a radiation pattern. If the frequencies differ slightly, the amplitude pulses at the difference frequency: that is the beat, and it is the principle used to tune an instrument and the one used by a superheterodyne receiver to shift a signal in frequency.

05Sound and the Doppler effect

Sound is a longitudinal pressure wave. Its intensity is measured on a logarithmic scale because the ear responds that way:

\[ \beta = 10 \, \log \frac{I}{I_0} \; [ \mathrm{dB} ] , \quad I_0 = {10}^{-12} \, W / m^2 \] The decibel of acoustics and that of electronics are the same definition: a ratio of powers on a logarithmic scale.

When the source or the receiver moves, the perceived frequency changes: this is the Doppler effect, f′ = f·(v ± v_obs)/(v ∓ v_source). Speed radar, the ultrasonic flowmeter and Doppler ultrasound imaging all work on it.

06The same thing, in electronics

Transmission lines. A cable that is long compared with λ does not behave like a conductor but like a medium that carries waves: there is reflection, standing waves and matching.
Signal integrity. On a fast digital board, an unterminated trace reflects the edge and produces steps and overshoot: it is a string with a free end.
Antennas. A half-wave antenna is, literally, a standing wave of current on a conductor.
Ultrasound. Distance sensors measure the time of flight of an acoustic pulse; flow sensors use the Doppler effect.
Crystals and SAW filters. They work with surface acoustic waves in a solid.

07In the lab

Activity 1 · Modes of a string

Stretch a thread between a vibrator and a pulley with weights. Vary the frequency until each mode appears and verify that f_n = n·v/(2L). Change the tension and check that v = √(T/μ).

Activity 2 · Speed of sound with a tube

With a tube partially submerged in water and a tuning fork, look for the resonance positions. The distance between two of them is λ/2, and with the fork’s f you obtain the speed of sound.

Activity 3 · Reflection in a coaxial cable

Send a pulse down a few meters of coax with the end open, short-circuited and terminated in 50 Ω, and look at the reflection on the oscilloscope. It is the activity that closes the circle: the three rows of the reflection table, measured.

08Common mistakes

Believing the wave carries matter. It carries energy; the medium only oscillates.
Thinking the frequency changes when the medium changes. λ changes, because v changes.
Confusing the propagation speed with the speed of a particle of the medium. They are different things and of very different orders of magnitude.
Believing sound travels faster if there is more pressure. In an ideal gas it depends on the temperature, not on the pressure.
Adding intensities in decibels. Decibels are not added: powers are added.
Miscounting the nodes: mode n has n antinodes and n+1 nodes including the ends.

09Self-assessment

A 1.2 m string with v = 120 m/s. What is its fundamental frequency?

λ₁ = 2L = 2.4 m, so f₁ = 120/2.4 = 50 Hz. The third harmonic is at 150 Hz.

What is λ for a 100 MHz signal in a coax with a velocity factor of 0.66?

v = 0.66·3·10⁸ ≈ 2·10⁸ m/s, so λ = 2·10⁸/10⁸ = 2 m. A quarter wave is 50 cm.

What happens to a wave when it is reflected at a fixed end?

It returns inverted, with a 180° phase shift. It is the case of the short-circuited line.

Two sounds of 440 and 443 Hz are played together. What is heard?

A tone of about 441.5 Hz whose intensity pulses 3 times per second: the beating used to tune an instrument.

If the power of a signal doubles, by how many decibels does it rise?

10·log(2) ≈ 3 dB, in both acoustics and electronics.

Why does a half-wave antenna resonate?

Because its open ends impose current nodes, and with L = λ/2 the reflected wave returns in phase: a standing wave is established, just as on the string.

10Further reading

A. P. French. Vibrations and Waves. W. W. Norton, 1971 (Spanish edition, “Vibraciones y ondas”, Reverté, 1974). The best book on the subject for an engineer: waves, modes and superposition from scratch.
Raymond A. Serway and John W. Jewett. Physics for Scientists and Engineers with Modern Physics, Volume 1. 9th ed. (Spanish edition, “Física para ciencias e ingeniería”), Cengage Learning, 2014. Chapters 16 to 18: mechanical waves, sound, superposition and standing waves.
David M. Pozar. Microwave Engineering. 4th ed., Wiley, 2011. Chapter 2 shows that the transmission line is this same wave equation.
Paul A. Tipler and Gene Mosca. Physics for Scientists and Engineers, Volume 1. 6th ed., W. H. Freeman, 2008 (Spanish edition, “Física para la ciencia y la tecnología”, Reverté, 2010). Good development of the Doppler effect and of the decibel scale.