A wave can have many properties that may interest us. For example, the frequency of a sound determines how high its pitch is. We can relate certain aspects (such as frequency, wavelength and velocity) of a wave by the following formulae: $$f=\frac{1}{T}$$ $$v=f\lambda$$ $$v=\frac{\lambda}{T}$$
where $$ \begin{equation}\begin{aligned} f&\rightarrow frequency\\ T&\rightarrow period\\ v&\rightarrow velocity\\ \lambda&\rightarrow wavelength\\ \end{aligned}\end{equation} $$
Mission details
- Find the frequency of a sound wave which has a period of $0.02\ s$. $$ \begin{equation}\begin{aligned} f=\frac{1}{T}=\frac{1}{0.02\ s}=50\ Hz\\ \end{aligned}\end{equation} $$
- Find the velocity of the same wave if its wavelength is $2\ m$. $$ \begin{equation}\begin{aligned} v&=f\times \lambda\\ &=50\color{orange}\ Hz\color{black}\times 2\ m\\ &=100\ m\color{orange}s^{-1}\\ \end{aligned}\end{equation} $$ Alternatively: $$ \begin{equation}\begin{aligned} v=\frac{\lambda}{T}=\frac{2\ m}{0.02\ s}=100\ ms^{-1}\\ \end{aligned}\end{equation} $$
- Determine the speed of a wave with wavelength $20\ m$, given that it completes $2$ oscillations in $4\ s$. The period is the time taken to complete $1$ oscillation and we are told it takes $4\ s$ to complete $2$ oscillations thus the period is: $$ \begin{equation}\begin{aligned} T=\frac{4\ s}{2\ oscillations}=2\ s\\ \end{aligned}\end{equation} $$
We do not write ‘oscillations’ as oscillation is dimensionless (e.g. the wheel made 2 oscillations)
Thus the speed is: $$ \begin{equation}\begin{aligned} v=\frac{\lambda}{T}=\frac{20\ m}{2\ s}=10\ ms^{-1}\\ \end{aligned}\end{equation} $$
Alternatively: The frequency is the number of oscillations in $1\ s$. $4$ oscillations occurred in $2\ s$ thus the number of oscillations in $1\ s$ would be: $$ \begin{equation}\begin{aligned} \frac{2\ oscillations}{4\ s}=0.5\ Hz\\ \end{aligned}\end{equation} $$ Thus the frequency is $0.5\ Hz$ and the speed will be: $$ \begin{equation}\begin{aligned} v&=f\times \lambda\\ &=0.5\ \color{orange}Hz\color{black}\times 20\ m\\ &=10\ m\color{orange}s^{-1}\\ \end{aligned}\end{equation} $$