$$ \begin{aligned} &h\,\frac{D}{Dt}(\zeta+f)-(\zeta+f)\frac{Dh}{Dt}=0 \\[6pt] &\mathrel{\underset{\text{expand }D/Dt,\;Dh/Dt}{\Rightarrow}}\; h\!\left(\frac{\partial\zeta}{\partial t}+u\frac{\partial\zeta}{\partial x}+v\frac{\partial\zeta}{\partial y}+\frac{Df}{Dt}\right) -(\zeta+f)\!\left(\frac{\partial h}{\partial t}+u\frac{\partial h}{\partial x}+v\frac{\partial h}{\partial y}\right)=0 \\[6pt] &\mathrel{\underset{f=f_0+\beta y,\;h=H+\eta}{\Rightarrow}}\; (H+\eta)\!\left(\frac{\partial\zeta}{\partial t}+u\frac{\partial\zeta}{\partial x}+v\frac{\partial\zeta}{\partial y}+\beta v\right) -(\zeta+f)\!\left(\frac{\partial\eta}{\partial t}+u\frac{\partial\eta}{\partial x}+v\frac{\partial\eta}{\partial y}\right)=0 \\[6pt] &\mathrel{\underset{\beta\text{-plane: replace }f\to f_0\text{ in factor}}{\Rightarrow}}\; (H+\eta)\!\left(\frac{\partial\zeta}{\partial t}+u\frac{\partial\zeta}{\partial x}+v\frac{\partial\zeta}{\partial y}+\beta v\right) -(\zeta+f_0)\!\left(\frac{\partial\eta}{\partial t}+u\frac{\partial\eta}{\partial x}+v\frac{\partial\eta}{\partial y}\right)=0 \\[6pt] &\mathrel{\underset{\text{linearize: keep only first order}}{\Rightarrow}}\; \Big(H+\cancel{\eta}\Big)\!\left(\frac{\partial\zeta}{\partial t} +\cancel{u\frac{\partial\zeta}{\partial x}} +\cancel{v\frac{\partial\zeta}{\partial y}} +\beta v\right) -\Big(\cancel{\zeta}+f_0\Big)\!\left(\frac{\partial\eta}{\partial t} +\cancel{u\frac{\partial\eta}{\partial x}} +\cancel{v\frac{\partial\eta}{\partial y}}\right)=0 \\[6pt] &\mathrel{\underset{\text{drop cancelled terms}}{\Rightarrow}}\; H\,\frac{\partial\zeta}{\partial t}+H\,\beta v-f_0\,\frac{\partial\eta}{\partial t}=0 \end{aligned} $$ ◀ ▶