Consider a sinusoidal mechanical wave travelling along the positive x-axis of a uniform string.
(a) Starting from the condition that a given phase of the wave (for example, a crest) is described by the equation kx−ωt=constant, derive an expression that relates the wave speed v to the angular frequency ω and the wave number k.
(b) Hence obtain the relation v=λν, clearly stating the intermediate steps and the definitions of all symbols used.
(c) When the same string is attached to a second string of different linear density, the wave is transmitted across the junction without any change in source. Using the relation derived in part (b), discuss which quantities among λ,ν,v will change at the boundary and which will remain the same. Give physical reasons for your answer.
(d) A common misconception is that increasing the amplitude of the wave will increase its speed. Using energy considerations or otherwise, refute this statement for waves on a stretched string.