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Find the number of natural transverse vibration of a string length l in the frequency interval from omega to omega +domega if the propagation velocity of vibrations is equal to v. All vibrations are supposed to occur in one plane. |
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Answer» Solution :Suppose the string is stretched along the `x` axis from `x=0 t o x=l` with the end points fixed. Suppose `y(x,t)` is the transverse DISPLACEMENT of the element at `x` at time `t`. Then `y(x,t)` obeys `(del^(2)y)/(delt^(2))=V^(2)(del^(2)y)/(delx^(2))` We look for a stationary of this equation `y(x,t)=A `sin`(omega)/(V) x sin (omegat+delta)` where `A &delta` are constants.. In this from `y=0 at x=0`. The further condition `y=0 at x=l` implies `(omega l)/(v)=NPI, N gt 0` `N` is the number of modes of frequency `le omega` Thus `dN=(l)/(piV)d omega` |
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