Waves

A wave is a disturbance that travels through a medium and carries energy from one place to another without the bulk transport of matter. Ripples and sound are examples. This chapter studies mechanical waves, which need a material medium. It distinguishes transverse waves, in which particles move perpendicular to the direction of travel, from longitudinal waves, in which they move along it with compressions and rarefactions. The displacement of a progressive wave is written y(x, t) = a sin(kx - omega t + phi), and the amplitude, wavelength, angular wave number, period, frequency and angular frequency are defined. You will derive the speed of a travelling wave, v = f lambda = omega/k, and study the speed of a transverse wave on a stretched string, v = sqrt(T/mu), and of sound, including Newton's formula and Laplace's correction. The principle of superposition explains the reflection of waves, standing waves, nodes, antinodes and the normal modes of strings and air columns. The chapter ends with beats, used to tune musical instruments.

Detailed Worksheet: Waves Section A - Definitions (10 marks) 1. Distinguish between transverse and longitudinal waves with one example each. (2 marks) 2. Find the wavelength of a sound of frequency 170 Hz in air, taking the speed of sound as 340 m/s. (2 marks) 3. Why can sound not travel through a vacuum? (2 marks) 4. What is the distance between two successive nodes in a standing wave? (2 marks) 5. Two tuning forks of 256 Hz and 260 Hz are sounded together. Find the beat frequency. (2 marks) Section B - Calculations and Applications (15 marks) 6. For the wave y = 0.05 sin(10x - 40t) in SI units, find the amplitude, wavelength, frequency and wave speed. (3 marks) 7. A string under a tension of 100 N has a mass per unit length of 0.01 kg/m. Find the speed of transverse waves on it, and the fundamental frequency if its length is 0.5 m. (3 marks) 8. Find the speed of sound in air at 27 degrees Celsius if it is 331 m/s at 0 degrees Celsius. (3 marks) 9. A pipe closed at one end is 0.25 m long. Find its fundamental frequency and the next possible harmonic. Take v = 340 m/s. (3 marks) 10. Explain why only odd harmonics are present in a pipe closed at one end. (3 marks) Section C - Diagrams (10 marks) 11. Draw the first three normal modes of a string fixed at both ends and mark the nodes and antinodes. (4 marks) 12. Draw the first two modes of vibration of an air column in a pipe closed at one end. (3 marks) 13. Draw a diagram of a longitudinal wave in a spring showing compressions, rarefactions and the wavelength. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Explain Newton's formula for the speed of sound in a gas and Laplace's correction. Calculate both values for air at STP, taking P = 1.013 x 10^5 Pa, density 1.29 kg/m^3 and gamma = 1.4. (5 marks) 15. Explain the formation of standing waves on a string fixed at both ends and derive the frequencies of its normal modes. (5 marks) 16. A pipe 30 cm long is open at both ends. Which harmonic of the pipe resonates with a 1.1 kHz source? Will resonance with the same source occur if one end is closed? Take v = 330 m/s. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Take the speed of sound in air as 340 m/s unless stated otherwise. Use SI units.