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Transverse Waves

{transversewaves.swf}

Transverse waves consist of particles vibrating up and down, \emph{thus} creating a wave in the left-right direction. 
The wave is essentially formed by a continuum of vibrating particles, each moving in simple harmonic motion. 
(You can 
think of the resultant left-right wave as being caused by a chain-effect of up and down particles affecting their neighbor's motion, 
as shown in the Flash movie above.)

\subsection{Transverse Wave Velocity}

While the individual particles propagate at rapidly changing velocities, the wave (\emph{formed by all the particles}) propagates 
at a velocity dependent on the tension ($T$) in the string and the mass density ($\mu$). This formula applies for small amplitude waves, 
which is valid for most practical applications. 

\begin{equation}
v=\sqrt{\frac{T}{\mu}}
\end{equation}

\includegraphics{transversewavevelocity.eps}

The diagram illuminates the similar triangles involved, where $v'$ is the vibrational velocity and $v$ is the wave velocity. 

\begin{eqnarray}
\frac{T}{T_y}&=&\frac{vt}{v't}\\
\frac{T t}{(\mu v t )v'}&=&\frac{v}{v'}\\
\frac{T}{\mu v v'}&=&\frac{v}{v'}\\
v^2&=&\sqrt{\frac{T}{\mu}}
\end{eqnarray} 

The second lines follows from impulse-momentum, noting that $T_{y} t = \Delta p =(\mu v t )v'$. 
Note that: $\mu=\frac{m}{l} \Rightarrow m=\mu vt$. The length ($l$) of the string of mass ($m$) is the distance propagated by the wave. 


Of course, the usual formula relating wavelength ($\lambda$) and frequency ($f=1/T$) for waves applies to transverse waves: 
\begin{equation}
v=\sqrt{\frac{T}{\mu}}=\lambda/T=\lambda f 
\end{equation}
 

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