\(\beta_{\max } \Rightarrow d _{\min } \text { and } \beta_{\min } \Rightarrow d _{\max }\)

\(d = d _{0}+ a _{0} \sin \omega t\)

\(d _{\max }= d _{0}+ a _{0} \text { and } d _{\min }= d _{0}- a _{0}\)

\(\therefore \beta_{\min }=\frac{\lambda D }{ d _{0}+ a _{0}}\) and

\(\therefore \beta_{\max }=\frac{\lambda D }{ d _{0}- a _{0}}\)

\(\beta_{\max }-\beta_{\min }=\frac{\lambda D }{ d _{0}- a _{0}}-\frac{\lambda D }{ d _{0}+ a _{0}}=\frac{2 \lambda Da _{0}}{ d _{0}^{2}- a _{0}^{2}}\)