Wave on a Graph

a) Look at the graphic for the the maximum and minimum point.

b) With the graphic, see the displacement and the time interval, and use the definition for the speed.

c) Look at the graphic for the difference with the tiles for two consecutive crests.

d) Since the velocity and the wavelength are known, the frequency can be directly calculated.

a) The amplitude is then the distance from \(y=0\) to a maximum (crest) or a minimum (valley), which is \(1\,\text{cm}\).

b) The displacement \(\Delta x\) is \(3/4\,\text{cm}\) and in a time interval of \(\Delta t=0.5\,\text{s}\). The definition of speed is:

\begin{equation*}
v=\frac{\Delta x}{\Delta t},
\end{equation*}

which numerically is

\begin{equation*}
v=\frac{3}{2}\,\text{cm/s}.
\end{equation*}

c) The distance between two crests. From the graph we see that the first crest in the black line is at tile 2 while the next crest is at tile 10. Then: \(8\times 3/4\,\text{cm}=6\,\text{cm}\). Hence the wavelength \(\lambda\) is \(6\,\text{cm}\).

d) Since \(v = \lambda f \), solving for the frequency and with numerical values we get:

\begin{equation*}
f=0.25\,\text{Hz}.
\end{equation*}

For a more detailed explanation of any of these steps, click on “Detailed Solution”.

a) The wave is clearly a sinusoidal wave moving along the X-axis and oscillating in the Y-axis. The amplitude is then the distance from \(y=0\) to a maximum (crest) or a minimum (valley). Noticing in the graph that this distance is \(1\,\text{cm}\) then the amplitude of the wave is \(1\,\text{cm}\).

b) Let us find now the speed of the wave. From the graph we notice that each division across the X axis is \(3/4\,\text{cm}\). Notice also that the whole wave moves one tile, that is the displacement \(\Delta x\) is \(3/4\,\text{cm}\) in a time interval of \(\Delta t=0.5\,\text{s}\). We can then use the definition for the speed \(v\) as

\begin{equation}
v=\frac{\Delta x}{\Delta t},
\end{equation}

which numerically is

\begin{equation}
v=\frac{3/4\,\text{cm}}{0.5\,\text{s}},
\end{equation}

\begin{equation}
v=\frac{3}{2}\,\text{cm/s}.
\end{equation}

c) To find the wavelength of the wave we must find the distance along the X axis in which the wave makes 1 oscillation. This is equivalent to find the distance between two crests or two valleys. From the graph we see that the first crest in the black line is at tile 2 while the next crest is at tile 10. Then there are 8 tiles of difference, which is equivalent to a distance of \(8\times 3/4\,\text{cm}=6\,\text{cm}\). Hence the wavelength \(\lambda\) is \(6\,\text{cm}\).

d) In order to find the frequency of the wave \(f\)we must use the relation between the wavelength, speed and frequency for any sinusoidal wave which reads

\begin{equation}
v=\lambda f.
\end{equation}

Solving for \(f\) we get

\begin{equation}
f=\frac{v}{\lambda},
\end{equation}

which after using the numerical values found before we get

\begin{equation}
f=\frac{3/2\,\text{cm/s}}{6\,\text{cm}},
\end{equation}

\begin{equation}
f=0.25\,\text{Hz}.
\end{equation}