1. Mechanical Waves Types of waves One-Dimensional Traveling Waves Wave speed in a stretched string Reflection and transmission of waves Energy Transmitted by sinusoidal Waves along a string
2. All waves carry energy, but the amount of energy transmitted through a medium and the mechanism of transport responsible differ from case to case. Eg. The power of ocean waves during a storm is much greater than the power of sound waves generated by a single human voice. All mechanical waves require: (1) some source of disturbance, (2) a medium that may be disturbed, (3) some physical mechanism through which elements of the medium can influence each other. MECHANICAL WAVES o They are governed by Newtons laws, and they can exist only within a material medium Eg. Water wave, wave on a string and sound wave
3. Eg. Sound waves. LONGITUDINAL WAVES The motion of the element of the wave (or medium) is parallel to the direction of travel of the wave Eg. Waves on a string and electromagnetic waves. TRANSVERSE WAVES The motion of the element of the wave (or medium) is perpendicular to the direction of travel of the wave TYPES OF WAVES MOTION:
4. The motion of water elements on the surface of deep water in which a wave is propagating is a combination of transverse and longitudinal displacements, with the result that elements at the surface move in nearly circular paths. Each element is displaced both horizontally and vertically from its equilibrium position. Q: How is the energy of an earthquake transported? http://resonanceswavesandfields.blogspot.com/2007/12/types-of-waves-i.html
5. Amplitude, A : the magnitude of the maximum displacement of the elements from their equilibrium positions as the wave passes through them. Wavelength, l : the distance between two successive identical points on the wave (eg. distance between two successive crests) Frequency, f : the number of complete cycles per unit time Period, T : the time required for 1 complete oscillation; T = 1/f Wave velocity, v : the velocity at which any part of the wave (eg. the crest) moves; v = fl Phase of the wave: the argument (kx wt) of the sine function. Phase constant f : The value of f chosen so that the function gives some other displacement and slope at x =0 when t = 0. WAVE CHARACTERISTICS
6. As if taking a still picture of a wave; amplitude and wavelength may be determined but not frequency and period. As if observing just a single point on a wave; amplitude and period may be determined but not wavelength . Representations
7. The wave equation can also be written as )sin( tAx w displacement amplitude angular frequency time )(cos tAy w Sine wave )(sin fw tAy phase angle or Displacement x(t) 0 -A A 1/4T 1/2T 3/4T T 3/2T x=A cos(wt +f) Amplitude Period T (second) Time (second) The equation of motion is sinusoidal as a function of time In this case, f =/2, x = A cos (wt - /2) or x= A sin wt Wave equation (Function of time)
8. ONE DIMENSIONAL TRAVELING WAVE Pulse traveling to the right Pulse traveling to the left The function y, sometimes called the wave function, depends on the two variables x and t.
9. vtxinAtxy l 2 s),( T tx inAtxy 2s),( )(s),( tkxinAtxy Phase xAxytAt l 2 sin)0,(,s0 k (= 2/l) is angular wave number and w (= 2/T) is angular frequency Consider a one dimensional wave traveling along the x-axis: Suppose the wave move to the right at velocity v. At t s later, the whole wave has moved to the right a distance, vt
10. The wave equation assumes that the displacement is zero at t = 0. However if the displacement is not zero at t = 0, we generally express the wave in the form: where f is the phase constant )(s),( f tkxinAtxy Eg. A sinusoidal wave traveling in the positive x direction has an amplitude of 15.0 cm, a wavelength of 40.0 cm, and a frequency of 8.00 Hz. The vertical displacement of the medium at t = 0 and x = 0 is also 15.0 cm, (a) Find the angular wave number k, period T, angular frequency, and speed v of the wave. (b) Determine the phase constant , and write a general expression for the wave function.
11. Ans; a) k = 2/l = 2 rad/ 40.0 cm = 0.157 rad/cm T =1/f = 1/(8.00 s-1) = 0.125 s w = 2f = 2(8.00 s-1) = 50.3 rad/s v =f l = (40.0 cm)(8.00 s-1) = 320 cm/s b) Because A = 15.0 cm and because y = 15.0 cm at x = 0 and t = 0, substitution into the wave function gives 15.0 = (15.0)sinf or sin f = 1 We may take the principal value f = /2 rad (or 90). Hence, the wave function is of the form y = Asin(kx wt + /2) = Acos(kx- wt) Substituting the values for A, k, and into this expression, we obtain y = (15.0 cm) cos(0.157x - 50.3t )
12. Example, Transverse Wave
13. Example, Transverse Wave, Transverse Velocity, and Acceleration
14. Linear wave equation in general: Refer Serway page 479 For waves on strings, y represents the vertical position of elements of the string. For sound waves propagating through a gas, y corresponds to longitudinal position of elements of the gas from equilibrium or variations in either the pressure or the density of the gas. In the case of electromagnetic waves, y corresponds to electric or magnetic field components.
15. Wave Speed on a Stretched String The speed of a wave along a stretched ideal string depends only on the tension and linear density of the string and not on the frequency of the wave. A small string element of length Dl within the pulse is an arc of a circle of radius R and subtending an angle 2q at the center of that circle. A force with a magnitude equal to the tension in the string, t, pulls tangentially on this element at each end. The horizontal components of these forces cancel, but the vertical components add to form a radial restoring force . For small angles, If m is the linear mass density of the string, and Dm the mass of the small element, The element has an acceleration: Therefore,
16. Eg. A uniform string has a mass of 0.300 kg and a length of 6.00 m. The string passes over a pulley and supports a 2.00-kg object. Find the speed of a pulse traveling along this string.
17. Reflection and Transmission (phase change) (no phase change)
18. For a sinusoidal wave of frequency f, the particles move in a SHM, and each particle has an energy For a 3-D elastic medium, assuming the entire medium has the uniform density r, Thus m =rV = rAl = rAvt; giving Therefore, energy transported by a wave is proportional to the square of the frequency and to the square of the amplitude. particleofmassismwhere, 2 1 m k fBecause 22'2 2 AvtfAE r Energy and Power of a Wave
19. = 2 ThereforeSince 22 2222' 2 1 2 1 2 1 Avt Avt l m AvtAE w wwr Power 22 2 1 Av t E P w Energy and Power Transmitted by sinusoidal waves on a string