Representation of amplitude as height of wave This may seem an odd question and i fully realize the futility of a possible change in representation (having participated in a gazillion such discussions in mathematics). That said ...
I am thinking that using the height of a wave to represent amplitude is a very poor representation. It gives the impression that with an increase in amplitude above 1, the particle is traveling further in the same amount of time, that is, that the speed of light increases with an increase in amplitude and vice-versa. Perhaps thickness or color of the wave would have been a better representation. Then it would have been a clearer that wavelength and frequency are directly related to the distance the particle travels along the wave and the formula lamda=v/f easy to explain.
(As I understand it, changing the amplitude changes the energy given to the slinky and not the length (wavelength) or stretch (frequency).)
 A: Strange as it may seem when the motion is simple harmonic the period and frequency of oscillation is independent of the amplitude.  
So for a simple pendulum doubling the amplitude of small oscillations does not change the period (frequency) of the simple pendulum.  
For mechanical waves doubling the amplitude does mean that the particles which are undergoing simple harmonic motion have to move a further distance in the same time and so have to travel faster.
The maximum speed of a particle undergoing simple harmonic motion at a frequency $f$ and amplitude $A$ is $2\pi f A$.
So doubling the amplitude doubles the maximum speed of a particle but does not change the speed of the wave.
For electromagnetic waves there are oscillations of electric fields and magnetic field and there is the speed of the wave.
If you choose a position you will find that at that position an electric field and a magnetic field at right angles to the electric field will change in magnitude and direction.
If the amplitude of the electric field is doubled so is the amplitude of the magnetic field and more power is transferred by the wave but the wave speed stays the same.
That doubling of the amplitude of a field does double the maximum rate of change of the field but that is not changing the speed at which the wave moves.
