1. PA 114 Waves and Quanta · Unit 2: Waves PA114 Waves and Quanta Unit 2: Waves (Problem solving) Tipler, Chapters 15 & 16 www.astro.le.ac.uk/~rda5/PA1140 Dr Richard Alexander (G44B)

2. PA 114 Waves and Quanta · Unit 2: Waves Problem solving lecture - Wave definitions - Standing waves and beats - Wave equation - Wave energy - Wave velocity

3. PA 114 Waves and Quanta · Unit 2: Waves The wave function for a wave on a string is given by: - In which direction does the wave travel? - What is the wave speed and wavelength? - What is the maximum transverse speed of an element of the string?

4. PA 114 Waves and Quanta · Unit 2: Waves Show that the superposition of two identical waves travelling in opposite directions is a standing wave

5. PA 114 Waves and Quanta · Unit 2: Waves Trig identities:

6. PA 114 Waves and Quanta · Unit 2: Waves Standing wave

7. PA 114 Waves and Quanta · Unit 2: Waves Show that the superposition of two waves with different frequencies results in beats

8. PA 114 Waves and Quanta · Unit 2: Waves Note heard at mean of the two frequencies, modulated (beats) at difference between freqs.

9. PA 114 Waves and Quanta · Unit 2: Waves A wave on a string has mass density  and is described by the wave function: (a)Show that this is a solution of the wave equation (b) By considering the stretching and transverse velocity of the string, show that energy propagates along the string at the wave speed.

10. PA 114 Waves and Quanta · Unit 2: Waves A piano wire is 1 m long, weighs 1 g, and is put under a tension of 40 N. Find the frequency of the first 3 harmonics of the wire. What tension would have to be placed on the wire to make the fundamental middle C (262 Hz)?