Lecture 2 Interference S3 S2 P Incident wave (wavelength l) y L d S1 Content Interference of Sound waves Two Slit Interference of Light Young Interference Phasors Multiple Slit Interference In[.]
Lecture 2: Interference P y S3 d S2 S1 Incident wave (wavelength l) L Content: Interference of Sound waves Two-Slit Interference of Light: Young Interference Phasors Multiple-Slit Interference Interference of Waves: When two waves are present at the same point in space and time (single w), they lead to interference: Add amplitudes (e.g., pressures or electric fields) What we observe however is Intensity (absorbed power) I = A2 Stereo speakers: Listener: y1 = A1cos(kx - wt) y2 = A1cos(kx - wt + ) y1 +y = 2A1 cos( / 2) cos(kx wt / 2) A = 0:waves add “in phase” (“constructive”) I = |2 A1|2 = 4|A1|2 = 4I1 = p:waves add “out of phase” (“destructive”) I = |2 A1*0|2 = Interference Exercise The relative phase of two waves also depends on the relative distances to the sources: The two waves at this point are “out of phase” Their phase difference depends on the path difference d r2 - r1 r1 Path difference d l/4 l/2 l Phase difference A= 2A1cos(/2) I Each fraction of a wavelength of path difference gives that fraction of 360º (or 2p) of phase difference: d = 2p l Solution The relative phase of two waves also depends on the relative distances to the sources: The two waves at this point are “out of phase” Their phase difference depends on the path difference d r2 - r1 r1 Path difference d I 4I1 Phase difference p/2 Each fraction of a wavelength of path difference gives that fraction of 360º (or 2p) of phase difference: Reminder: A can be negative “Amplitude” is the absolute value A = 2A1cos(/2) 2A1 2A1 2I1 l/4 p 0 l/2 2p 2A1 4I1 l d = 2p l Amplitude vs Intensity (for interfering waves) cos(/2) cos2(/2) Plot here as a function of A = 2A1cos(/2) 2A1 Constructive Interference I = 4A12cos2(/2) 4A12 Destructive Interference 0 2p l 4p 2l 6p 3l 8p 4l Note: What is the average intensity? 10p 5l d Iave = 4I1*0.5 = 2I1 Sound wave example: Each speaker alone produces intensity I1 = W/m2 at the listener, and f = 900 Hz Drive the speakers in phase Compute the intensity I at the listener: Sound velocity in air: v = 330 m/s d = 2p l r1 = 2p(d/l) 3m m 1) 2) 3) 4) 5) Procedure: Compute path-length difference: d = r2 - r1 = Compute wavelength: l = Compute phase difference (in degrees): = Write a formula for the resultant amplitude: Compute the resultant intensity, I = with d = r2 – r1 Sound wave example: Each speaker alone produces intensity I1 = W/m2 at the listener, and f = 900 Hz Drive the speakers in phase Compute the intensity I at the listener: Sound velocity: v = 330 m/s d = 2p l r1 = 2p(d/l) 3m m 1) 2) 3) 4) 5) with d = r2 – r1 Procedure: Compute path-length difference: d = r2 - r1 = m Compute wavelength: l = v/f = (330 m/s)/(900 Hz) = 0.367 m Compute phase difference (in degrees): = 360 (d/l) = 360(1/0.367) =981 Write a formula for the resultant amplitude: A = 2A1cos(/2), A1 = I1 Compute the resultant intensity, I = I1cos2(/2) = (1 W/m2 ) (0.655)2 =1.72W/m2 Exercise : Speaker interference r1 What happens to the intensity at the listener if we decrease the frequency f? (Recall the phase shift was 981.) a decrease b stay the same c increase Exercise : Speaker interference r1 What happens to the intensity at the listener if we decrease the frequency f? (Recall the phase shift was 981.) a decrease b stay the same I c increase = 981 f decreases l increases d/l decreases 360 decreases I decreases (see figure) 720 900 Summary The resultant intensity of two equal-intensity waves at the same point in space is: I = I1cos2(/2) For nonequal intensities, the maximum and minimum intensities are Imax = |A1 + A2|2 Imin = |A1 - A2|2 The phase difference between the two waves may be due to a difference in their source phases or a difference in the path lengths to the observer In the latter case: = 2p(d/l) with d = r2 – r1 .. .Interference of Waves: When two waves are present at the same point in space and time (single w), they lead to interference: Add amplitudes (e.g.,... cos2(/2) Plot here as a function of A = 2A1cos(/2) 2A1 Constructive Interference I = 4A12cos2(/2) 4A12 Destructive Interference 0 2p l 4p 2l 6p 3l 8p 4l Note: What is the average intensity?... Speaker interference r1 What happens to the intensity at the listener if we decrease the frequency f? (Recall the phase shift was 981.) a decrease b stay the same c increase Exercise : Speaker interference