Waves: Class 11 Physics

Key points

A wave is a disturbance that carries energy from one place to another without carrying matter with it. Mechanical waves need a medium, such as water waves, sound and waves on a string. Electromagnetic waves, such as light, do not need a medium. In a transverse wave the particles vibrate at right angles to the direction of travel (crests and troughs). In a longitudinal wave they vibrate along the direction of travel (compressions and rarefactions). Sound in air is longitudinal.

Wave terms: the wavelength λ is the distance between two successive crests (or two successive compressions). The frequency f is the number of waves passing a point per second. The period is T = 1/f. The amplitude is the maximum displacement of a particle. All these combine in the wave equation v = fλ, where v is the speed of the wave. The frequency is fixed by the source, and the speed is fixed by the medium.

The speed of a transverse wave on a stretched string is v = √(T/μ), where T is the tension and μ is the mass per unit length. A tighter or lighter string carries waves faster. The speed of a longitudinal wave in a fluid is v = √(B/ρ), where B is the bulk modulus. Newton thought that sound travels in air with constant temperature (isothermal) changes, which gave a value of about 280 m/s. Laplace corrected this by pointing out that the compressions are so quick that they are adiabatic, and the formula became v = √(γP/ρ). It gives about 332 m/s at 0°C, in agreement with experiment.

Factors affecting the speed of sound in air:

• Temperature: the speed increases with temperature, about 0.6 m/s for each degree Celsius, so v ≈ 331 + 0.6t m/s, or about 343 m/s at 20°C.
• Density: at the same temperature, sound travels faster in a gas of lower density.
• Pressure: no effect, if the temperature stays the same, because pressure and density change together.
• Humidity: moist air is less dense, so the speed is slightly higher.
• Speed is much higher in solids and liquids than in gases.

The principle of superposition says that when two or more waves meet at a point, the resultant displacement is the vector sum of the individual displacements. After they pass each other, the waves continue unchanged. Interference is the result. In constructive interference the waves meet in phase (crest on crest), and the amplitude increases. In destructive interference they meet out of phase by 180° (crest on trough), and the amplitude decreases. The path difference for constructive interference is a whole number of wavelengths, nλ. For destructive interference it is an odd number of half wavelengths, (n + ½)λ.

Stationary (standing) waves are formed when two identical waves travel in opposite directions, for example an incident wave and its reflection. Points that stay at rest are called nodes and points of maximum vibration are called antinodes. The distance between two successive nodes is λ/2. Unlike a travelling wave, a stationary wave does not transfer energy along the medium.

Stationary waves on a string fixed at both ends of length L:

• Nodes form at both ends. The allowed wavelengths are λ = 2L/n, where n = 1, 2, 3 and so on.
• The frequencies are f = n v/2L = (n/2L)√(T/μ).
• The lowest frequency (n = 1) is the fundamental or first harmonic, f₁ = v/2L. Higher frequencies are 2f₁, 3f₁ and so on.
• Guitar and sitar strings work in this way: pressing a string shortens L and raises the pitch.

Stationary waves in air columns:

• An open pipe has antinodes at both ends. The frequencies are f = nv/2L for n = 1, 2, 3 and so on, so all harmonics are present.
• A closed pipe (closed at one end) has a node at the closed end and an antinode at the open end. The frequencies are f = nv/4L for odd n only (1, 3, 5), so only odd harmonics are present.
• The fundamental of a closed pipe is half that of an open pipe of the same length.

Beats are the periodic rise and fall in loudness heard when two sounds of slightly different frequencies f₁ and f₂ are sounded together. The number of beats per second equals the difference of the frequencies, f_beat = |f₁ − f₂|. Beats are used by musicians to tune instruments. When the beat frequency becomes zero, the two notes are in tune.

The Doppler effect is the apparent change in the frequency of a wave when the source and the observer are in relative motion. When the source moves towards a stationary observer, f' = f v/(v − v_s), so the pitch rises. When it moves away, f' = f v/(v + v_s), and the pitch falls. The change in the sound of a passing ambulance or train is a common example. The effect is also used in police radar, in medical ultrasound to measure blood flow, and by astronomers, where the red shift of light shows that distant galaxies are moving away.

Study tip: Draw the node and antinode pattern of the pipe or string before you write the length in terms of wavelength. This avoids most mistakes in harmonics. For the Newton and Laplace question, write both formulas and say why the process is adiabatic.

Definitions and flashcards

Say the meaning first, then open the card to check.

Formula sheet

Print this page

Short questions with model answers

These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.

Numericals with worked steps

Common mistakes

• Thinking that a wave carries matter along with it. Only energy is transferred.
• Using the fundamental frequency formula of an open pipe for a closed pipe, or the other way round.
• Saying that the speed of sound depends on pressure at a fixed temperature. It does not.
• Changing the frequency when a wave enters a new medium. Frequency stays the same, while speed and wavelength change.
• Reversing the signs in the Doppler formula. An approaching source gives a higher frequency, so v_s is subtracted in the denominator.

Practise this chapter

Practice questions and timed test Printable worksheet