1. 1. S Standing Waves in Music Christopher Cheng 32929144
2. 2. What is a Standing Wave? S 2 harmonic waves with equal amplitude, wavelength and frequency moving in opposite directions S When they collide, each segment of the string oscillates in SHM S Can be represented by the equation S D(X,t)=2A(sin(kx)cos(wt) S And amplitude at a given point can be detemined by S A(x)=2Asin(2pi(x/wavelength)
3. 3. Parts of a Standing Wave S Nodes: occus at points where A(x)=zero S Antinodes: Occur where the wave has maximum amplitude of 2A S As you can see from the graph the distance between two nodes is half a wavelength S And distance between a node and its consecutive antinode is qaurter of a wavelenth
4. 4. Fundamental Frequency S Also referred to as first harmonic S Lowest Frequency at which the string will oscillate S Can be determined by eqn
5. 5. Fundamental Frequency (cont) S F2=2f1 S f3=3f1 S and so on.
6. 6. For an explanation of standing waves watch from 1:10- 2:00
7. 7. Standing Waves in Instruments S The formation of standing waves is whats responsible for producing musical notes S Eg. Flutes, sound waves propagate and when they reach the end, they reflect back, creating notes. S But sounds that are created by instruments are the result of many different frequencies.
8. 8. Question S A Guitar String has a linear mass density of 4.5x10-4 kg/m and has a tension of 80N. It takes 8.7x10-4 s for the string to travel from one end to the other S What is the wavespeed of the string? S How long is the string? S What is the frequency of the first three Harmonic
9. 9. Solution S What is the wavespeed of the string? S V=(T/u)(1/2) S V=(80/4.5x10-4)0.5=421.6 m/s
10. 10. Solution #2 S Delta t = (L/V) S 8.7x104s=(L/421.61 m/s) S L= 0.36m
11. 11. Solution #3 S Using the equation on the right and substituting values we find that the fundamental frequency is 585.6hz S Second harmonic= 2f1= 1171hz S Third harmonic=3f1=1756.8hz