Title 15.Waves.pdf ✅

NEET-2022 Ultimate Crash Course PHYSICS Waves
POINTS TO REMEMBER 1. Some of the waves In nature are neither transverse nor longitudinal, but a combination of the two. Waves on the surface of water (known as ripples) are an example. When water wave travels on the surface of deep water, water molecules at the surface move up and down; and back and forth simultaneously describing nearly circular paths as shown in figure The surface of water has been shown in the form of crests and troughs. As the wave passes, water molecules at the crests move In the direction of the wave while those at the troughs move in the opposite direction
1. In case 0  = / 2, from equation y x t A kx t ( , sin ) = − + (  0 ) y x t A kx t ( , sin / 2 ) = − + (   ) or y x t A kx t ( , cos ) = − (  ) and in case 0  = , y x t A kx t ( , sin ) = − + (  0 ) y x t A kx t ( , sin ) = − + (   ) = + − = − − A kx t A kx t sin sin      ( ) ( )   or y x t A t kx ( , sin ) = − ( ) 2. At any given time t, equation y x t A kx t ( , sin ) = − (  ) gives the displacement of the particle as a function of coordinate x. At t = 0, 2 y A kx A x sin sin     = =     3. At any given coordinate x (i.e., position) equation y x t A kx t ( , sin ) = − (  ) given the displacement of the particle as a function of time t. At x= 0, ( ) 2 y A t A t A t sin sin sin T  = − = − = −   4. The displacement y of a particle depends both on x and t, i.e., y is a function of two independent variables x and t. A graph of y x t A kx t ( , sin ) = − (  ) would be a surface if we plotted x and t along rectangular axes in a horizontal plane and plotted y vertically above the plane. 5. It is important to note that wave velocity (also called phase velocity) v is different from the particle velocity (vp) at a distance x from the origin. As y A kx t = − sin (  ), p sin cos ( ) ( ) ( ) dy d v A kx t A kx t dt dt = = − = − −            or v A kx t p = − −   cos( ) Thus, we find that particle velocity ( p v ) is a varying quantity as it depends on space (x) and time (t), whereas wave velocity (v) is constant for a given medium. Further, amplitude of particle velocity = A 6. Slope of the wave cos( ) dy kA kx t dx = = − ( ) ( ) cos / cos p v A kx t v dy dx kA kx t k     − − − = = = − − Thus, v v dy dx p = − ( / ) Hence, particle velocity at a given position = − wave v velocity  slope of the right going wave at that point. In case the wave is going left, v v dy dt p = ( / ) 7. Particle acceleration, cos( ) p dv d a A kx t dt dt = = − −       or a A kx t = − − − −      sin ( ) ( )   or ( ) 2 a A t kx = + −   sin Clearly, acceleration amplitude 2 = A 8. Hydrogen (H2) and Oxygen (O2), both being diatomic gases, have the same value of  . Further, oxygen is 16 times heavier than hydrogen. Thus, from a b v T   or a b v kT = 
16 4 H O O H v v   = = = or 4 H O v v = Thus, sound waves travel four times faster in hydrogen than these do in oxygen. 9. The speed of sound in helium at 0°C is 972 m/s, which is much greater than its value in air at 0°C, i.e., 331 m/s. An interesting and amusing demonstration of this fact is the variation in the human voice when the vocal cavities are partially filled with helium. If a person talks before and after taking a deep breath of helium, the result is a high-pitched sound. The increase in frequency corresponds to an increase in the speed of sound in helium. 10. All wave motions can he divided into two groups : (i) Electromagnetic waves (ii) Mechanical waves. 11. There is another type of waves, called matter waves which are associated with moving electrons, protons and other fundamental particles. 12. With the exception of waves on the surface of a liquid, all other mechanical waves are called elastic waves, 13. Electromagnetic waves require no medium to sustain them whereas mechanical waves need a medium, to sustain them. 14. Though in everyday language, the concept of sound is related to, hearing, the elastic waves are also called sound. 15. Mechanical waves can he transverse, longitudinal or a combination of the two, 16. Pressure waves are those elastic waves which are generated due to variations in pressure in the gaseous medium and are longitudinal waves. .Sound is the most common example this type of waves. 17. The waves set up in fluids (liquids and gases) are always longitudinal. 18. The waves set up in solids can he either transverse or longitudinal. 19. The mechanical waves set up in a string under tension are transverse waves. 20. In a vibrating tuning fork, the waves in the stem are longitudinal while those in the prongs are transverse. 21. Torsional waves are those which are propagated along a rod when one of its ends is given a rapid twist while other is clamped in this case, the displacement of the particles is neither parallel nor transverse to the axis of the rod but around the axis of the rod (without changing its shape). 22. In a wave motion, energy and momentum are transferred or propagated. 23. A mathematical expression of the form: y x t f x vt ( , ) =  ( ) is the equation of a travelling or progressive wave. 24. 2 2 2 2 2 d y d y v dt dx = represents a wave motion propagated with a definite velocity v. 25. Particle velocity (vp ) and wave velocity (v) are not the same. 26. As wave number, k = 2 /   , its value depends on (a) nature of the medium and (h) frequency of the source producing waves. 27. Whereas v k  = is called phase velocity, g d v dk  = ,is called the group velocity. 28. In a non-dispersive media (i.e., a medium in which all harmonic waves of different wavelengths travel with the carne speed), g v v = . But in a dispersive medium, g v may be larger or smaller than v. Sound waves in air are non-dispersive while light waves in glass are dispersive. It is due to this reason that light is split (or dispersed) into various colours while passing through a prism. 29. Ripples formed on the surface of a liquid are a combination of transverse and longitudinal waves although transverse waves are predominant. These waves are unique in the sense that their velocity of propagation depends upon their wavelength 30. Seismic waves produced during earthquakes consist of transverse waves (shear waves) as well as longitudinal waves (pressure waves). 31. Sound waves include not only the audible range but all mechanical waves as well. But to avoid confusion, we hose considered those waves which provoke nervous response constituting the hearing as sound.