1

13 Outline

• vibrations, waves, resonance

• Homework:

• 1, 2, 15, 30, 41, 45, 51, 64, 67, 101.

2

vibrations

• Examples:

• vibrating reed, mass on spring,

• drum, piano wire, string,…

• most vibrations are sinusoidal in time,

• and called “simple harmonic” motions (shm)

3

terminology• x: displacement

• A: maximum displacement

• f: frequency (cycles/s)

• angular frequency : (rad/s)

• k: spring constant (N/m)

4

sinusoidal nature of vibrations

5

Kinematics of SHM

• position:

• frequency:

• angular frequency:

• maximums:

)cos( tAx

Av max2

max Aa

m

kf

2

1

m

k

6

Dynamics of SHM

• F = -kx, a = -kx/m

• E = ½kx2 + ½mv2.

• Ex: k = 10N/m, m = 200grams, A = 10cm.

7

Waves

• traveling disturbance

• Transverse Longitudinal

8

wave phenomena

• interference of waves

• Examples, noise cancellation headphones, standing waves

• reflection, refraction, and diffraction.

9

Periodic Waves

• continuous, well defined amplitude (A), frequency (f), wavelength (), and speed v = f.

• Example: f = 10 hertz, = 3 m.

• v = (10/s)(3m) = 30m/s.

10

Waves on Strings

• Wave Velocity depends on:

• tension in string (F) and,

• the mass per unit length of string.

• Example: F=36N, m/L=0.010kg/m

Lm

Fv

/

smv /603600010.0

36

11

Standing Waves

• Nodes (places with zero amplitude)

• Anti-nodes (places with maximum amplitude)

…waves in which the amplitude at a given location does not vary with time. Due to wave interference.

Features:

12

Standing Wave: Both Ends Fixed

13

Standing Wave: One End Fixed, One End Free

14

summary

• many vibrations are simple harmonic

• one equation set describes all shm

• wave speed equations

• interference of waves & standing waves

• reflection, refraction, diffraction.

15

Main ResultskxF

Av max

)cos( tAx T

2

2max Aa

m

k

2

f

221 kxPEelastic

2212

21 kxmvEmech

16

Nature of Sound Waves

• Longitudinal

• Oscillations are:

• Condensations (higher pressure areas) and

• Rarefactions (lower pressure areas)

• Sound travels at about 343m/s at room temperature and normal atmospheric pressure

17

Doppler Effect

• Frequency received is different than the Source frequency due to:

• Source Motion,

• Receiver Motion or,

• a combination of Source and Receiver motions.

18

Example of wavelength distortion due to source motion:

19

Height vs. Time

0

0.5

1

1.5

2

2.5

0 5 10 15 20 25

Time (s)

Y (

m)

values of “A” and “f”?

20

Decibels

• intensity level

where Io = 1.0x10-12 W/m2.

• Example: Intensity of sound is 4.0x10-5 W/m2. Intensity level is

oI

IdB log10

dBdBdB 76100.4log10100.1

100.4log10 7

12

5

21

Sound Intensity (I)

• Intensity = power/area = P/A [watt/meter2]

• Spherical Radiation I = P/4r2.

• Example: Small speaker emits 1.0W of sound in all directions. Intensity 10m from the speaker is 1.0/(4102) W/m2.

22

Frequency of Sound

• Audible Range: 20Hz to 20,000Hz

• Infrasonic: f < 20Hz

• Ultrasonic: f > 20,000Hz

23

sinusoidal nature of shm

• position of blue mass moving on spring turns out to be same as the horizontal position of an object in uniform circular motion.

)cos( tAx

24

L

gf 2 small angles

Simple Pendulum