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Unformatted text preview: Oscillations & Waves Oscillations Topics of Discussion Topics Types of Waves A Mathematical Description of Waves Principle of Linear Superposition Standing Waves Standing Waves on a String The Speed of a Wave on a String Two Types of Waves Two When oscillations occur along the When direction of propagation, we call the wave a longitudinal wave. longitudinal When oscillations occur in a plane When perpendicular to the direction of propagation, we call the wave a transverse wave. transverse The Mathematical Description The of a Wave
When a wave travels from one location to When another, we will call such a wave a traveling wave. This type of wave is described by wave This y ( x, t ) = ym sin ( ω t − kx )
where ym is the amplitude, x is the position, ω is the angular frequency, t is time, and ωt – kx is the phase of the wave. the Principle of Superposition Principle
When N sources are simultaneously When present at the same place, the resulting disturbance is the sum of the disturbances from the individual waves, that is from y ( x, t ) = ∑ N i= 1 ym , i sin ( ω i t − ki x ) . Standing Waves Standing
A standing wave is the superposition of two waves traveling in opposite directions. It is of the form the y ( x, t ) = y1 ( x, t ) + y 2 ( x, t )
= 2 y m sin kx cos ω t = ym sin ( ω t − kx ) + ym sin ( ω t + kx ) where ω is the angular frequency, ym is the where maximum amplitude and kx-ωt is the phase. kx- Standing Waves on a String Standing The ends of the string are fixed. All points on the string oscillate with the same All frequency. frequency. Different points on the string have different Different amplitudes (or displacements). amplitudes Nodes are locations where NO motion occurs. Anti-Nodes are locations of maximum amplitude (or displacement). amplitude The distance d between nodes is d=λ/2 where d= The λ is the wavelength of the wave. Standing Waves on a String cont. Standing The type of waves traveling on a string are The transverse waves. transverse The (linear) natural or fundamental frequency The or occurs when one anti-node is present, and is given by f1=v/2L, where v is the speed of the where wave and L is the length of the string. An integer multiple n of the natural frequency An is called the nth-harmonic wave, and is given and by fn=nv/2L=nf1 where n=2,3,4,.... n=2,3,4, The Speed of a Wave on a String The
Let the length and the mass of a string be Let L and m, respectively. If the applied force respectively. acting on a string is F then the velocity of the wave is given by the v= F . mL ...
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- Spring '10