Sound and Standing Waves

• Intensity
• depends on distance - how?
• consider a source radiating energy at a fixed rate, uniformly in all directions
• the intensity is the power per unit area.
• as we get farther from the source, the energy is spread over a larger area
• the area is proportional to the square of the distance from source



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• because the average power is determined by the source



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• Doppler Effect

• the perceived pitch depends on both the motion of the source and the motion of the observer
• Case I
• observer moving, source stationary
• speed of waves relative to observer is v' = v + v_o
• observer moving to source
• the new frequency will be



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• if the observer is moving away from the source, +v_o replaced by -v_o
• Case II
• source moving, observer stationary
• wavelength changes



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• if the source is moving away from the observer, -v_s is replaced by +v_s



• Both source and observer moving



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• the frequency will increase if source and observer are approaching each other
• the frequency will decrease if source and observer are receding from each other

Standing Waves on a String

• consider a string fixed on both ends.
• disturbances created in the string will be reflected at either end
• we will have waves traveling in both directions on the string
• these waves will interfere. (See Chapter 16) and produce "standing waves"
  
• at certain select frequencies
• the waves will interfere in such a way that certain points on the string will not oscillate or will have an amplitude of oscillation equal to zero
• points are called nodes.
• Other points on the string will experience a maximum oscillation
• points are called anti-nodes
• All points on the string will oscillate with the same frequency.
• The distance between two nodes or two anti-nodes is one-half wavelength.
• The distance between a node and an anti-node is one-quarter wavelength.
• Only certain frequencies will produce standing waves
• Determining those frequencies
• Condition: We must have a node at each end.
• The lowest possible frequency will occur when we produce a standing wave with the longest possible wavelength.
• the longest possible wavelength that satisfies the condition that both ends are nodes is shown below.

• the wavelength of this wave is equal to 2L



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• The next longest wavelength




• this wavelength is equal to L



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• the next longest wavelength


• this wavelength is equal to 2/3L
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• the trend will be



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