Category : JEE Main & Advanced
(1) If during the propagation of a progressive wave, the particles of the medium perform SHM about their mean position, then the waves is known as a harmonic progressive wave.
(2) Suppose a plane simple harmonic wave travels from the origin along the positive direction of x-axis from left to right as shown in the figure.
The displacement y of a particle 1 at O from its mean position at any time t is given by \[y=a\sin \omega \,t.\]
The wave reaches the particle 2 after time \[t=\frac{x}{v}\]. Hence displacement y of a particle 2 is given by
\[y=a\sin \omega \,\left( t-\frac{x}{v} \right)\]\[=a\sin (\omega \,t-kx)\] \[\left( \because \,k=\frac{\omega }{v} \right)\]
The general equation of a plane progressive wave with initial phase is
(3) Various forms of progressive wave function.
(i) \[y=a\,\sin \,(\theta t-kx)\]
(ii) \[y=a\,\sin \,(\omega t-\frac{2\pi }{\lambda }x)\]
(iii) \[y=a\sin 2\pi \left[ \frac{t}{T}-\frac{x}{\lambda } \right]\]
(iv) \[y=a\sin \frac{2\pi }{T}\left( t-x\frac{T}{\lambda } \right)\]
(v) \[y=a\sin \frac{2\pi }{\lambda }(vt-x)\]
(vi) y = \[a\sin \omega \,\left( t-\frac{x}{v} \right)\]
(4) Particle velocity : The rate of change of displacement y w.r.t. time t is known as particle velocity
Hence from \[y=a\sin \,(\omega \,t-kx)\]
Particle velocity \[{{v}_{p}}=\frac{\partial y}{\partial t}=a\omega \,\cos (\omega \,t-kx)\]
Maximum particle velocity \[{{({{v}_{p}})}_{\max }}=a\omega \]
Also \[\frac{\partial y}{\partial t}=-\frac{\omega }{k}\times \frac{\partial y}{\partial x}\]\[\Rightarrow \]\[{{\upsilon }_{p}}=-v\times \] Slope of wave at that point
(5) Important relations for numerical solving
(i) Angular frequency \[\omega =\] co-efficient of \[t\]
(ii) Propagation constant \[k=\] co-efficient of \[x\]
Wave speed \[v=\frac{\text{co-efficient of }t}{\text{co-efficient of }x}=\frac{\omega }{k}\]
(iii) Wave length \[\lambda =\frac{\text{co-efficient of }x}{2\pi }\]
(iv) Time period \[T=\frac{2\pi }{\text{co-efficient of }t}\]
(v) Frequency \[n=\frac{\text{co-efficient of }t}{2\pi }\]
(vi) \[{{({{v}_{p}})}_{\max }}=a\omega =a(2\pi n)=\frac{a2\pi }{T}\]
(vii) If the sign between t and x terms is negative the wave is propagating along positive X-axis and if the sign is positive then the wave moves in negative X-axis direction.
(viii) Co-efficient of sin or cos functions i.e. Argument of sin or cos function is represented by phase i.e. \[(\omega t-kx)=\]Phase.
(ix) Phase difference and path difference : At any instant t, if \[{{\phi }_{1}}\] and \[{{\phi }_{2}}\] are the phases of two particles whose distances from the origin are \[{{x}_{1}}\] and \[{{x}_{2}}\] respectively then \[{{\varphi }_{1}}=(\omega \,t-k{{x}_{1}})\] and \[{{\varphi }_{2}}=(\omega \,t-k{{x}_{2}})\]\[\Rightarrow \]\[{{\varphi }_{1}}-{{\varphi }_{2}}=k({{x}_{2}}-{{x}_{1}})\]
\[\Rightarrow \]Phase difference \[(\Delta \varphi )=\frac{2\pi }{\lambda }\text{.}\,\text{Path difference (}\Delta x)\] \[(x)\]
(x) Phase difference and time difference : If the phases of a particle distance \[x\] from the origin is \[{{\phi }_{1}}\] at time \[t\] and \[{{\phi }_{2}}\] at time \[{{t}_{2}},\] then \[{{\varphi }_{1}}=(\omega \,{{t}_{1}}-kx)\] and \[{{\varphi }_{1}}=(\omega {{t}_{2}}-kx)\]\[\Rightarrow \] \[{{\varphi }_{1}}-{{\varphi }_{2}}=\omega ({{t}_{1}}-{{t}_{2}})\]
\[\Rightarrow \]Phase difference \[(\Delta \varphi )=\frac{2\pi }{T}.\,\text{Time difference}\,\text{(}\Delta t)\]
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