SPH 3U1

Unit 4: Waves and Sound

Note 5: Standing Waves & Resonance

Reference: Chapter 9.2, 9.4, 10.4

Demo: Students make standing waves.

When waves of equal amplitude and wavelength traveling in opposite directions interfere, a standing wave interference pattern can be set up.

Nodes - are the points that remain at rest (i.e. displacement is zero, destructive interference)

Antinodes - are where there is maximum constructive interference (i.e. supercrests and supertroughs)

A standing wave pattern is a stationary pattern of successive nodes and anitnodes.

Internodal Distance - the distance between successive nodes. It is 1/2 wavelength.

Picture.....

Example 1: A standing wave occurs in a lake when Susan dips her paddle repeatedly in the water between two barriers. The nodes are every 100 cm.

a) What wavelength of wave is she making?

b) If the wave speed in the lake water is 0.80 m/s, how often does the paddle dip?

Example 2: A standing wave has a distance of 80 cm between five consecutive nodes.

a) What is the wavelength of the wave?

b) What is the speed of the wave if the source frequency is 20 Hz?

STRINGED INSTRUMENTS:  Standing Waves Between 2 Fixed/ Closed Ends

• A vibrating string with 2 fixed ends will form a standing wave pattern. In the simplest mode of vibration the string vibrates in only one segment called the fundamental frequency.
• Understand pictures/harmonics/overtones (See Table 1 pg423)

AIR COLUMNS:

Open Air Columns - Standing Waves Between 2 Free Ends

• Have an antinode at the open end. Show why this makes sense (ie. based on their knowledge of reflections at open ends).
• Draw an equivalent Table to Table 1 for this situation.

Closed Air Columns - Standing Waves Between 1 Free End and 1 Fixed End

• Have a node at the fixed end. Show why this makes sense
• Draw an equivalent Table to Table 1 for this situation

Example 3: The speed of a wave on a string with two fixed ends is 340. m/s. The frequency of the wave is 220. Hz. What length of string is required to make a standing wave with the 2nd harmonic?

Example 4: The speed of a wave on a string with a fixed end and a free end is 340. m/s. The frequency of the wave is 220. Hz. What length of string is required to make a standing wave with the 2nd harmonic?

Example 5: The first resonant length of a closed air column occurs when the length is 18 cm. What is the wavelength of the sound? If the frequency is 512 Hz, what is the speed of sound? (Ans. 72 cm, 369 m/s)

Example 6: If the frequency in a closed air column is 256 Hz and the speed of sound is 344 m/s, what is the second resonant length? (Ans. 1.01 m)

Example 7: An organ pipe 3.60 m long, open at both ends, produces a musical note at its fundamental frequency. What is the wavelength of the note produced? What is the frequency? (Ans. 7.20 m, 47.8 Hz)

Resonance and Damping:

Resonance

The frequency at which a medium vibrates mot easily.

Damping

The anplitude of a wave decreases because of energy absorption or destructive interference. (i.e. swing or pendulum's amplitude decreasing to a stop).

If the energy put into the swing or pendulum equals the energy lost as a result of friction (air resistance) then the amplitude remains constant. If the energy is greater, the object can keeping swinging higher and higher.

Mechanical Resonance:
(Think of a swinging on a swing)
Mechanical resonance is the vibrating response of an object to a periodic force from a source that has the same frequency as the natural frequeny of the object.
Examples: Swings, Rocking a car to get out of the snow, Tacoma Narrows Bridge
Show Video of Tacoma Narrows Bridge

Earthquakes can causes vibrations in buildings. To decrease the amplitude of these vibrations some buildings have a mas damper usually made of a concrete or steel pendulum.

Acoustic Resonance:
It's responsible for producing the sound in musical instruments.
http://physics.decapoa.altervista.org/fisica/index.php?w0=resonance&&id=47

Pg 425 #1,2,3

Pg 426 #2-8

Complete Handout 'Natural Frequencies of Vibration for Columns of Air'