If we were to consider a spring stretched across a surface as a medium for a wave pulse, then can we expect the amplitude of a transverse pulse effect the speed of the wave?
The relationship which defines the speed of a wave in a medium is
S = LF where S is the speed, L is the wavelength and F is the frequency.
For a spring pulse, the frequency is not an easily measured quantity directly (how do you get the frequency of just one event?), so it can be replaced by a relationship to the tension and mass of the spring. See first reference below for equations.
The result is that the wave speed equation becomes
S = Square Root (k T/d) where k is the spring constant in newtons/m, T is the tension in the spring in newtons, and d is the linear density of the spring in kg/m.
We can see in this equation that there are no terms or variables that relate to the amplitude (height, or strength) of the pulse.
Therefore, the wave speed does not depend upon the magnitude of the pulse.