Superposition

Principle of SuperpositionInterference

Stationary Waves

What happens when two waves occupy the same space at the same

time?

Principle of Superposition• When two waves occupy the same

space at the same time, their displacements add up (algebraically.)

• Interference

Constructive Interference• Amplitudes increase

Destructive Interference• Amplitudes decrease• Sometimes, they completely cancel each

other out

Interference and Path Difference

• Consider two sources of waves at a distance from each other.

• What conditions are necessary for complete destructive interference to occur at a point?

• What conditions are necessary for complete destructive interference to occur at a point?

• The same intensity or amplitude• The same type of wave• They have the same frequency or period

• What conditions are necessary for complete destructive interference to occur at a point?

• They must be in phase• Their path difference must be

• N (λ/2)• Where N = 1, 3,5,….

• What should be their path difference for constructive interference to occur at a point?

• Their path difference must be• N (λ)• Where N = 1, 2,3,….

Example 1• Two loudspeakers A and B emit are driven

by the same amplifier and emit in phase waves.

Example 1• For what frequencies does constructive

interference occur at P?

Example 1• For what frequencies does destructive

interference occur at P?

Example 2• Two identical sources of sound are

situated 80 cm apart as shown.

• They always vibrate in phase with each other.

Example 2• A microphone is placed 100 cm from S1 as

shown. The speed of sound is 330 ms-1.

• How many frequencies between 1.0 kHz to 4.0 kHz will produce no sound at M?

Standing Waves (a.k.a. Stationary Waves)

Formation of Standing Waves

• Incident waves travel in one direction and are reflected back

• Incident and reflected waves interfere• Points that constructively interfere form

antinodes• Points that destructively interfere form

nodes

Standing Waves (Which is a standing wave?)

Do standing waves have wave speed?

• Yes• Wave speed is the speed of the incident

and reflected waves.

Equations for Standing Waves

• Fixed Ends– String Instruments– ƒ = n (v/2L)

• Where n = 1,2,3,…

Equations for Standing Waves• One fixed end, One open end

– Percussion instruments– Water pipe– ƒ = n (v/4L)

• Where n = 1,3,5,…

Equations for Standing Waves

• Two Open ends– Some wind instruments– ƒ = n (v/2L)

• Where n = 1,2,3,…• Similar to two fixed ends

Example 1• One of the 63.5-cm long strings of a

guitar is tuned to produce the note B3 (frequency 245 Hz) when vibrating in its fundamental mode. Find the speed of the transverse waves on the string.

Example 1• If the speed of sound in the air is 344

ms-1, find the frequency and wavelength of the sound wave produced in the air by the vibrating string.

Example 2 (A Tale of Two Pipes)

• One day when the speed of sound is 345 ms-1, the fundamental frequency of a stopped organ pipe is 220 Hz. How long was this pipe?

Example 2 (A Tale of Two Pipes)

• The second overtone of this pipe has the same wavelength as the third harmonic of an open pipe. How long is the open pipe?