Initialization

SetOptions[Plot,PlotStyle->{Red,AbsoluteThickness[2]},
ImageSize->500,AxesStyle->Directive[FontFamily->"Arial",FontSize->18,Black,AbsoluteThickness[0.5],Arrowheads[0.04]]];

Work

BASIC CONCEPTS IN WAVES : "MATERIAL" VELOCITY 

1 - D Waves, Oregon State University, Department of Physics. Janet Tate, 1997 - 2012

The name "material velocity" is rather restrictive, and has physical meaning only when the wave propagates in a material like string or air. It then refers to the velocity of the particles that support the disturbance.  The concept, though, can also be applied to fields (e.g. E, B) that propagate in vacuum.

Start with a waveform (say a transverse wave on a string) that has wavelength 1 m, period 1 s, and amplitude 1 m.

ψ[x_,t_]:=1*Cos[2*Pi*x-2*Pi*t]

Now isolate the displacement of the string at the position x = 0:

g[t_]:=ψ[0,t]

Look at these functions over a distance correpsonding to a little over one wavelength and a time of two periods. The waveform  is presented in red, and blue marks the position x = 0.

Animate[Plot[ψ[x,t],{x,-1,1},
PlotStyle->{Red,Thick},
AxesLabel->{x,ψ},
PlotLabel-> t, 
Epilog->{Blue,Disk[{0,g[t]},{.02,.03}]}],
{t,0,2},AnimationRunning->False]

1.  Pick any particular feature like a crest or a trough or a zero.  
How far does the feature move in the horizontal (x) direction in, say, 1 s?  
What is the feature's velocity in m/s?
2.  Concentrate on the blue portion of the string at x = 0.  
How far does the blue piece of string move in, say, 1 s?  
What is the velocity of the blue portion of the string in meters/second?  
3.  Does the blue portion of the string move in the horizonal direction at all?
4.  Is the difference between the "phase velocity" (or feature velocity) in the horizontal direction and the "string velocity" or "material velocity" or "transverse velocity" (the velocity of any part of the string) clear to you?  If not, ask an instructor.
5. How would you distinguish between the two in the case of a longitudinal wave?