1 The speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of a wave travelling along the surface of a sea is believed to depend on
the depth of the sea, \(d \mathrm {~m}\),
the density of the water, \(\rho \mathrm { kg } \mathrm { m } ^ { - 3 }\),
the acceleration due to gravity, \(g\), and
a dimensionless constant, \(k\)
so that $$v = k d ^ { \alpha } \rho ^ { \beta } g ^ { \gamma }$$ where \(\alpha , \beta\) and \(\gamma\) are constants.
By using dimensional analysis, show that \(\beta = 0\) and find the values of \(\alpha\) and \(\gamma\).