Chapter 6: Waves

The chapter problem: the concert hall check

A hall is being prepared for a concert with a sound and light show. The technician has to check the equipment before the audience arrives.

These are the facts to use all the way through the chapter.

• The two main loudspeakers are 1.20 m apart. A signal generator drives them in phase at 1.00 kHz. The speed of sound in the hall is 340 m s⁻¹.
• The light show uses a red laser of wavelength 633 nm.
• The laser beam passes through a double slit whose two slits are 0.25 mm apart, and the pattern falls on a screen 2.50 m away.
• The same beam also falls on a diffraction grating ruled with 300 lines per millimetre.
• A polarising filter sets the plane of polarisation of the beam, and a second filter is turned until its axis is at 60° to the first.
• A guitar string is fixed at both ends. It is 0.650 m long, its mass per unit length is 4.0 × 10⁻⁴ kg m⁻¹, and its tension is 81 N. It is meant to sound the note E₄, which has a frequency of 330 Hz.

By the end of the chapter you will find the wavelength of the sound and the angle of the first loud band. You will find the spacing of the light fringes and the angles of the grating maxima. You will work out the wave speed and the lowest frequency of the guitar string. Then you will decide whether the string is tuned to E₄ and what the technician must change.

First contact: no working yet. Predict. The two loudspeakers play the same note, so in some places the sound is loud and in others it is quiet. Suppose the technician changes the note to a higher one. Do the loud and quiet places stay where they are, or do they move? Write down your answer and one sentence of reasoning.

By the end of this chapter you will be able to

• describe a progressive wave, and tell a transverse wave from a longitudinal wave;
• use $v = f\lambda$ and $T = \dfrac{1}{f}$;
• find the phase difference between two points on a wave;
• state the principle of superposition, and explain constructive and destructive interference;
• use path difference to decide whether two waves reinforce or cancel;
• explain what makes two sources coherent;
• use $w = \dfrac{\lambda D}{s}$ for the fringes from two slits;
• describe diffraction, and explain how the size of a gap changes the pattern;
• use $d\sin\theta = n\lambda$ for a diffraction grating, and find the highest order it can give;
• explain what polarisation is and why only transverse waves can show it;
• describe what a polarising filter does to light, and what happens when two are crossed;
• explain how a stationary wave forms, and define a node and an antinode;
• find the harmonics of a string and of an air column open at one or both ends;
• use $v = \sqrt{\dfrac{T}{\mu}}$ for a wave on a string.

Where it is used

A sound engineer places the speakers at a concert and uses interference to understand why some seats sound better than others. Noise-cancelling headphones superpose a wave on the noise so the two cancel. A spectroscopist uses a diffraction grating to split starlight into its wavelengths, and so find out what a star is made of. Polarising sunglasses cut the glare that reflects from water and roads. A guitar maker chooses the string's length, tension and thickness to set its note. Medical ultrasound and radio antennas both depend on the same wave ideas.

Your turn to connect it

Think of a time you noticed a wave effect. You could have heard sound change as you walked past loudspeakers, or seen colours in a soap film, an oil slick or the surface of a CD. Write down two examples. For one of them, say what you think caused the pattern.

Roadmap

1. 6.1 Superposition and interference
2. 6.2 Diffraction and polarisation
3. 6.3 Stationary waves
4. Review