Oscillations and Waves

Pendulum Harmonic wave with only one frequency N W

Mass on a spring Harmonic wave with only one frequency

Displacement vs time Displaced systems oscillate around stable equil. points amplitude Equil. point period (=T)

Simple harmonic motion Pure Sine-like curve T Equil. point T= period = time for 1 complete oscillation = 1/T f = frequency = # of oscillations/time

Masses on springs Animations courtesy of Dr. Dan Russell, Kettering University

Not all oscillations are nice Sine curves Equil. point T f=1/T

Natural frequency f= (1/2p)k/m f= (1/2p)g/l

Driven oscillators natural freq. = f0 f = 0.4f0 f = 1.1f0 f = 1.6f0

Mechanical Resonance Mechanical resonance is the tendency of a mechanicalsystem to respond at greater amplitude when the frequency of its oscillations matches the system's natural frequency of vibration (its resonance frequency or resonant frequency) than it does at other frequencies. The system responded as the frequency of its oscillations matched the natural frequency of vibration due to the wind.

Resonance in a wave Resonance - The increase in amplitude of oscillation of an electric or mechanical system exposed to a periodic force whose frequency is equal or very close to the natural undamped frequency of the system.

Waves Animations courtesy of Dr. Dan Russell, Kettering University

Wave in a string Animations courtesy of Dr. Dan Russell, Kettering University

Pulsed Sound Wave

Harmonic sound wave

Harmonic sound wave

V=fl or f=V/ l Harmonic wave =v =l l T = = fl = but 1/T=f distance Wave speed =v Shake end of string up & down with SHM period = T wavelength =l l T distance time wavelength period Wave speed = v = = = fl = V=fl or f=V/ l but 1/T=f

Reflection (from a fixed end) Animations courtesy of Dr. Dan Russell, Kettering University

Reflection (from a loose end) Animations courtesy of Dr. Dan Russell, Kettering University

Adding waves pulsed waves Animations courtesy of Dr. Dan Russell, Kettering University

Two waves in same direction with slightly different frequencies Adding waves Two waves in same direction with slightly different frequencies Wave 1 Wave 2 resultant wave “Beats” Animations courtesy of Dr. Dan Russell, Kettering University

Adding waves harmonic waves in opposite directions incident wave reflected wave resultant wave (standing wave) Animations courtesy of Dr. Dan Russell, Kettering University

Confined waves Only waves with wavelengths that just fit in survive (all others cancel themselves out)

Harmonics For every given frequency, there are other frequencies that perfectly fit in the same space. These are called harmonics. They are whole divisions of the original wavelength, or multiples of the basic frequency. This is well known in musical instruments.

Allowed frequencies l= 2L f0=V/l = V/2L f1=V/l = V/L=2f0 l=L l=(2/3)L Fundamental tone f1=V/l = V/L=2f0 l=L 1st overtone l=(2/3)L f2=V/l=V/(2/3)L=3f0 2nd overtone l=L/2 f3=V/l=V/(1/2)L=4f0 3rd overtone l=(2/5)L f4=V/l=V/(2/5)L=5f0 4th overtone

Ukuleles, etc l0 = L/2; f0 = V/2L l1= L; f1 = V/L =2f0 l2= 2L/3; f2 = 3f0 L l3= L/2; f3 = 4f0 Etc… (V depends on the Tension & thickness Of the string)

Standing waves in a closed tube

Standing waves in an open pipe

Doppler effect

Sound wave stationary source Wavelength same in all directions

Sound wave moving source Wavelength in forward direction is shorter (frequency is higher) Wavelength in backward direction is longer (frequency is higher)

Waves from a stationary source Wavelength same in all directions

Waves from a moving source Wavelength in backward direction is longer (frequency is higher) Wavelength in forward direction is shorter (frequency is higher)