Harmonic Waves

Physics 101

Overview

• Waves that undergo simple harmonic motion • Can be modeled using sine or cosine functions• Characteristics: – Amplitude = maximum displacement from

equilibrium (A)– Wavelength = length of one wave (λ)– Period = time needed to complete one cycle – Frequency = number of wave cycles/second (f )– Phase Constant = displacement at t=0 (Φ)

Wave Diagram

y = Amplitudeλ = Wavelength

Equations

1. Wave number = k = 2π/λ 2. Wave speed = λf = (ω/k)3. Wave function:

- Decreasing x: D(x,t) = Asin(kx+ωt+Φ)- Increasing x: D(x,t)= Asin(kx-ωt+Φ)

4. Instantaneous velocity = (-ωA)cos(kxo-ωt+Φ)

5. Instantaneous acceleration = -ω2Asin(kxo-ωt+Φ)

A 90cm long cello string has a frequency of 400Hz and a wavelength of 180cm. What is the speed of the wave?

Speed = Frequency x WavelengthSpeed = 400Hz x 1.8m = 720m/s* Always double check that the units cancel properly. Wavelength in this question is given in cm so it must be converted to meters before solving the question. *

A harmonic wave on a guitar string can modeled by the function D(x,t) = 1.2sin(5x-2t). Calculate the instantaneous velocity at t=1s at the point x=4 on the string.

Substitute t=1 and x=4 into the velocity function (derivative of displacement):

-2.4cos(5(4)-2(1)) = v(4,1) = -1.58m/s

Using the information from the previous question, calculate instantaneous acceleration at the same point.

Substitute t=1 and x=4 into the acceleration function (derivative of velocity):

4.8sin(5(4)-2(1)= a(4,1) = -3.6m/s2

*Notice that the instantaneous acceleration is proportional to the displacement:

a(4,1) = -ω2D(4,1) = -3.6m/s2