Lecture #4 Traveling Waves: Longitudinal versus Transverse Waves PHYS102

1. Lecture #4Traveling Waves: Longitudinal versus Transverse WavesPHYS102 January, 27th

2. Today’s Agenda 1) Transverse versus longitudinal waves; 2) How do we express a wave in mathematical form? 3) Introduction to Diffraction

3. Lecture objectives  1)Summary on Waves; Math form for simplest waves. We will focus on the simplest of all wave: The traveling wave. 2) Examples…examples…(the best way to understand and learn)

4. 1. Traveling waves: waves that travel (move, propagate) Longitudinal wave: the vibration of the particles of the medium is in a direction along the wave motion Transverse wave: the vibration of the particles of the medium is in a direction

5. Space dependence: Fix time t. Vary the location x along a straight line. We call x the propagation direction of the wave. For light, we say that x is the direction of the ray. Then, the wave is periodic, for a traveling wave. The repeat-distance is the wavelength. Its symbol is λ. Mathematical form, fixed time t. The value of the size of the peak (relative to the zero-value) is called the amplitude.

6. Pure Wave • Wavelength () disturbance  position

7. Pure Wave • Amplitude (A) disturbance position A

8. The precise form for the wave isWave value y = A sin (2πx/λ).Replacement of sine by cosine also expresses a wave. This replacement simply shifts the wave along the x direction by (1/4)λ.

9. Cosine moves the wave with /4 disturbance /4 position A

10. The angle inside the sine has units of radians. Reminder: One radian is that angle for which the arc length and radial length are equal. Check that this form is right: If we add the full wavelength λ to x, then the angle inside the sine simply increases by 2π. That is, by 360 degrees. Conclude: The wave-value is unchanged, as expected.

11. Time dependence: Now, fix the location x. Vary the time t. Then, the time dependence is also sinusoidal. The form for the t-dependence; Wave value = A sin (2πt/T). T is the period. It is the repeat-time. The frequency f of the wave is the number of full cycles per time. So, it is the inverse of T: f = 1/T.

12. The full form for the wave: Now, let both x and t vary. The full form for the wave is: Wave value = y = A sin [(2πx/λ) – (2πt/T)]= A sin [2π(x/λ – t/T)] This is the complete form for any traveling wave. It simply combines the previous two forms.

13. Wave speed v Follow the motion of the crests. The speed of their motion defines the wave-speed v. The speed v is given by the wave-relation v = fλ.

14. 3) Introduction to Diffraction Observation: Suppose light passes through an opening that is large, compared to its wavelength λ. Then, we find: The shadow it casts is sharp. It appears that there is no spreading out of the light

15. But: Suppose light passes through a tiny opening. We observe: The light spreads out like a fan. There is no sharp shadow. There is a bright area that fades into darkness. There are no sharp edges.  

16. Second observation: If we look more carefully, we can see alternate dark and bright fringes at the edges. On close examination, even the sharpest shadow is blurred slightly at the edges. These phenomena define diffraction.

17. Cause of DiffractionIts cause is interference.Light rays intersect the opening at differing points (call them A, B, C, etc.) within the opening. Pick any point P on the screen.The path-length of the ray from opening to P depends on the location(A, B, C, …..) of the point it intersects the opening.

18. Example: Suppose the rays from A and B (both to P) have lengths that differ by a multiple of λ.Then, they interfere constructively, reinforcing each other. But suppose the rays from A and C differ by (1/2)λ. Then there is destructive interference. (cancellation). For some points P on the screen, constructive interference dominates. Then, we see a bright spot. For other points on the screen, destructive interference dominates. Then, wee see a dark spot.  

19. This is the explanation of the alternating bright and dark fringes.