WAVES A wave is the propagation of a disturbance through a medium without any net displacement of the medium.
A wave causes the transfer of energy from one point in a medium to another without the actual transfer of matter between the points. WAVES
mechanical
Electromagnetic
Matter
Mechanical waves require a medium for transmission. Examples are water sound and seismic waves. Electromagnetic waves require no medium for transmission. Examples are light waves and radio waves. Matter waves are waves associated with the motion of electrons protons and other fundamental particles. Mechanical waves are classified as: (i) Longitudinal. (ii) Transverse Longitudinal Wave A longitudinal wave is a wave in which the particles of the medium oscillate with simple harmonic motion parallel to the direction of propagation of the wave. Sound wave is the only known longitudinal wave. Transverse Wave Particles of the medium execute simple harmonic motion in a direction perpendicular to the direction of propagation of the wave. CHARACTERISTIC OF A SIMPLE WAVE
Displacement - Distance from equilibrium position. Amplitude - Maximum value of the displacement. Wavelength Distance between any two successive corresponding point on the wave. Period Time taken for one complete oscillation. Frequency Number of waves passing a particular point per second. Phase difference Two waves are in phase if they reach their maximum amplitude at the same time, are zero at the same time and have their maximum amplitudes at the same time. Two waves are out of phase if they reach their maximum amplitude at different times are zero at different times. Phase difference can also be expressed in terms of wave length. Figure 1 shows two waves in phase while figure 2 shows two waves out of phase by
Figure 1 Waves in phase
Figure 2 Waves out of phase by
MATHEMATICAL REPRESNTATION OF A WAVE To completely describe a wave we need a function that gives the shape of the wave for a sinusoidal wave traveling the displacement is given by Y(x t) = YmSin ( kx wt + ) The sign indicates the direction that the wave is moving, for a wave moving towards the right the negative sign is used so the equation becomes Y(x t) = YmSin ( kx - wt + ) Similarly for a wave traveling towards the left the equation is Y(x t) = YmSin ( kx + wt + ) K is the angular wave number. The S.I. unit is radian per mete 2 W is the angular frequency and is given by: 2 2 w= w= T T k=
F is the frequency and is given by: 1 w f = = T 2 V is the wave speed and is given by: w V= k EXAMPLE 1 A particular wave is given by Y = (0.200 m) Sin[(0.500 m-1)x (8.20 rad/s)t] Find (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) Wave number. Amplitude. Wavelength. Angular frequency. Frequency. Period. Velocity of the wave. The displacement of the wave at x =10 cm and t = 0.500 s.
SOLUTION (i) (ii) (iii) (iv) (v) (vi) 2 = 0.500 rad /m A = 0.200 m k=
=
2 2 = = 12 .6m k 0.500
w = 8.20 rad/ s
f =
T =
w 8.20 = = 1.31 Hz 2 2
1 1 = = 0.766 s f 1.31
(vii) V =
w 8.20 = = 16 .4m / s k 0.500 V = f = (1.31)(12 .6) =16 .4m / s
Wave is moving towards the right since the sign in the wave is negative. (viii) Y = (0.200m) Sin [ ( 0.500 m-1)(10.0 m) (8.20 rad/s)(0.500s)] = (0.200 m) Sin[0.900] rad = (0.200 m)(0.783) = 0.157 m SPEED OF TRANSVERSE WAVE ON A STRING The speed of a transverse wave on a string is given by 3
