Standing Wave and Resonance

May 19th 2009

Standing Wave:

• Interference of two similar wave forms produces a standing wave.

Video on Standing Wave:• http://www.youtube.com/watch?v=MT7EpS4OX3k&feature=related• http://www.youtube.com/v/MT7EpS4OX3k&feature=related

Diagram 1:

• node: point that remains at rest.

antinode: point midway between the nodes where maximum constructive interference occurs.

• inter-nodal distance: it is the distance between successive nodes. It is equivalent to one-half the wavelength of the wave source.

dn=0.5λ

Ex.1: (Think-Pair-Share)

• a.) A standing wave occurs in a pond. The nodes are at every 38cm. What is the wavelength of wave?

• b.)What is the frequency of the wave if speed of the wave in the pond is 0.95m/s?

Solution

• a.) dn=0.5λ • 38=0.5λ • 38/0.5=λ • 76cm= λ • b.)v=fλ• f=v/λ• Note: 76cm is equal to 0.76m• f=(0.95m/s)/(0.76m)=1.25Hz• Thus, the frequency is 1.25Hz.

Ex.2: (Think-Pair-Share)

A standing wave has a distance of 45cm between four consecutive nodes. What is the wavelength of the wave? What is the speed of the wave in the medium if the frequency of the source is 30Hz?

Solution

• L=45cm• f=30Hz

λ=?• v=?• L=3 dn=3(0.5)λ• L=3dn=1.5λ• 45=1.5λ• 45/1.5=λ• 30=λ• λ=30cm• v=fλ• f=30Hz

λ=30cm=0.3m• v=(30)(0.3)=9m/s• v=9m/s

14.5: Resonance

• Definitions:fundamental frequency or first harmonic: Lowest frequency of a string denoted fo

• harmonics and overtones: Integer multiples of the fundamental frequency. For example, 2 fo, 3 fo, etc.

Diagram 2:

Mechanical Resonance

• Mechanical Resonance: : is the vibrating response of an object to a periodic force from a source that has the same frequency as the natural frequency of the object.

Video: Mechanical Resonance in the Tacoma Narrows Bridge

(Washington State, 1940)• http://www.youtube.com/watch?v=3mclp9QmCGs

•http://www.youtube.com/v/3mclp9QmCGs

Question: What caused the bridge to collapse?

Answer:

• Answer: The bridge had a design flaw that gave it a natural frequency that was coincidentally similar to the frequency of the wind gusts at that particular spot. Mechanical resonance caused large amplitude oscillations. Thus, the bridge collapsed.

Example 2 of mechanical resonance: Pushing a child on a swing.

• Push a child on a swing at the correct frequency, always at the same point in the cycle, and the child’s amplitude of swinging increases dramatically.

Wind Instruments:

• Wind Instruments: These instruments are nothing more than elaborate air columns in which a standing wave is formed.

Diagram 3:Closed air column on the left, open air column on the right.

Example 3:(Think-Pair-Share)

• Ex.3: A closed air column resonates at two consecutive lengths of 94.0cm and 156cm. If the speed is 350m/s, what is the resonant frequency of the air column?

• (Note: You’ll have to know how to do these calculations for tomorrow’s lab)(Hint: The difference in length between any two resonant lengths is always 0.5λ.)

Solution• L1= 94cm• L2=156cm• v=350m/s• The difference in length between any two resonant lengths is always 0.5λ.

0.5λ=L2-L1=156cm-94cm=62cm or 0.62m• 0.5λ=0.62m

λ=0.62/0.5

λ=1.24m

Lastly,

f= _v_ = 350m/s = 282Hz λ 1.24m

• Thus, the resonant frequency of the column is 282Hz

Pitch

• Pitch: Pitch is the highness or lowness of a musical note. A higher pitch has a higher frequency.Diagram 4:

Loudness

• Loudness: Loudness is related to the amplitude of the sound wave.

• Diagram 5:

Homework:

Homework: pg. 512 to 513

#19, 21, 22, 23, & 26