1. Consider the possible standing waves that could exist on a 6.00-m long stretched rope (fixed at both ends). a. What is the wavelength of the fourth harmonic frequency? b. If the fundamental frequency is 4.00 Hz., what is the speed of waves that travel along this string? 1.a. 2.00 mb. 72.0 m/s 2.a. 3.00 mb. 48.0 m/s 3.a. 4.00 mb. 36.0 m/s 4.a. 6.00 mb. 24.0 m/s 5.None of the above. 3/2/161Oregon State University PH 212, Class #26

2. What is the wavelength of the second lowest frequency standing sound wave in a tube of length L that has one end open and the other closed? 1.4L 2.2L 3.L 4.4L/3 5.2L/3 3/2/162Oregon State University PH 212, Class #26

3. The wavelength of the third harmonic on a stretched string is 10.0 cm while the frequency of the fourth harmonic is 480 Hz. What is the speed of the waves on this string? 1.12.0 m/s 2.36.0 m/s 3.48.0 m/s 4.60.0 m/s 5.120 m/s 3/2/163Oregon State University PH 212, Class #26 Solution: 3(0.10/2) = L and 480 = 4v/(2L)

4. 3/2/16Oregon State University PH 212, Class #264 More about Sound Sound waves are longitudinal waves. What’s oscillating? Several attributes, actually: Position, speed, acceleration. And pressure. Like the other properties, the pressure of the air (or whatever solid or fluid the sound is traveling through) varies sinusoidally along the direction of the wave’s travel: P = Acos[(2  / )x ± 2  ft] As with other waves, v sound can be computed via the equation v = f. But we can also predict v from the characteristics of the material. In general, sound travels faster in hotter gases. Why? And, sound travels faster in liquids than in gases (and faster in solids than in liquids). Why?

5. 3/2/16Oregon State University PH 212, Class #265 We experience sound (biologically) somewhat differently than we measure it (physically). MeasureExperience Pressure variation (Pa) Sound Frequency (Hz.) Pitch Absolute Intensity (W/m 2 ) Relative Intensity (dB) Loudness

6. 3/2/16Oregon State University PH 212, Class #266 Sound as a Pressure Variation The wave equation for sound can be expressed as a pressure variation—in units of pressure. For example, a wave equation for a typical level of speech would be something like this: P = (3x10 -2 Pa)cos[(2  / )x ± 2  ft] Notice how small the amplitude of that pressure variation is, compared to, say, atmospheric pressure— and yet the ear can detect this easily!

7. 3/2/16Oregon State University PH 212, Class #267 Frequency vs. Pitch Musical pitch and frequency are very closely related. On the musical scale, for example, each octave represents a doubling of frequency. And the frequency intervals within an octave that sound most “in tune” to our ears (those that form pleasant- sounding chords) are in whole-number ratios. For example, the tones in a major triad chord, such as C-E-G (or A, C#, E), have frequencies in the ratio of 4:5:6.

8. 3/2/16Oregon State University PH 212, Class #268 Absolute Sound Intensity All waves transmit energy. Sound transmits its energy via the pressure/displacement variations in the medium. The rate at which this energy is transmitted is the power of the sound— related to the amplitude and the frequency in the wave equation. Think of the work needed per second to start wave impulses. And so what energy arrives on a sound wave at a given point? What is the sound intensity, I, at a given location? Convention- ally, we measure the power (Watts) it transmits perpendicularly through a unit surface area (1 m 2 ) at that location: I = P/A. So if a point source with a power P emits sound in all directions (i.e. in a spherical pattern), it’s easy to calculate the absolute intensity, I, at any distance r from the source: I = P/4  r 2

9. 3/2/16Oregon State University PH 212, Class #269 Definition of absolute intensity: I = P/A I for a point source: I = P/4  r 2 Example: What is the absolute intensity of sound, at a distance of 10 m from a single speaker producing a total of 400 W of sound? 1.About 1/3 W/m 2 2.About 1 W/m 2 3.About 5/3 W/m 2 4.About 3 W/m 2 5.None of the above. Follow-ups: What is the absolute intensity of sound at a distance of 20 m from the above speaker? 30 m?

10. 3/2/16Oregon State University PH 212, Class #2610 Relative Sound Intensity Our ears don’t respond to sound consistently. For example, if the absolute intensity (as measured in W/m 2 ) is doubled, we don’t hear the sound twice as loud. To hear it twice as loud, we need about ten times the intensity—ten times the sound pressure power transmitting through 1 m 2. So our experience of loudness relates to absolute sound intensity on a logarithmic scale. We notice a change of loudness in proportion to the ratio of the intensities, I 2 /I 1, not the difference (I 2 – I 1 ). Thus, for measures of hearing, we use a logarithmic scale of relative intensity, , in units of decibels (dB). We compute the  of a given sound by comparing its absolute intensity I (in a ratio) to a reference sound, I 0 :  = 10 log(I/I 0 )

11. 3/2/16Oregon State University PH 212, Class #2611  = 10 log(I/I 0 ) The reference intensity, I 0 = 1 x 10 -12 W/m 2, is the faintest sound the average human ear can detect. Then, for every 10-dB increment in relative intensity (i.e. a 10-fold multiplication in absolute intensity, I), we hear about twice the loudness. (a) What is the dB rating of a sound with absolute intensity I = 1.0 x 10 -11 W/m 2 (e.g. rustling leaves)? (b) What is the dB rating of a sound with absolute intensity I = 1.0 x 10 -10 W/m 2 (e.g. a whisper)? 1. (a) –10 dB(b) –20 dB 2.(a) 1 dB(b) 11 dB 3.(a) 2.3 dB(b) 12.3 dB 4.(a) 10 dB(b) 20 dB 5.None of the above. So how much louder to our ears is a whisper than a rustle of leaves? About twice as loud.

12. 3/2/16Oregon State University PH 212, Class #2612 How much louder to our ears is a 73-dB sound than a 43-dB sound? 1.3 times as loud 2.6 times as loud 3.8 times as loud 4.30 times as loud 5.None of the above.

13. 3/2/16Oregon State University PH 212, Class #2613 Q: Every 10-dB increase in loudness is a ten-fold (10x) increase in intensity. So, is a 5-dB increase a five-fold (5x) increase in intensity? A: Find out.… Suppose: Sound 1 has a loudness of 40 dB. Sound 2 has a loudness of 45 dB. Find I 2 /I 1.