IMAGINE a PROPAGATED WAVE to be a sequence of (circular or spherical) ripples, each of which is created at the moment a dancer's toe touches a point in the plane.

AS THE DANCER MOVES on a PLANE, each of his landings sets of a single ripple with fixed speed v. Once the ripple center is created by the dancer, it remains fixed in space while its generated ripple continues to widen with a radius expanding at fixed speed v.

REGARDLESS of the DANCER'S SPEED, therefore, the speed of each ripple that is generated by the toe of the dancer does NOT DEPEND ON THE SPEED OF THE DANCER'S TOE.

EXAMPLE: Sound waves heard from a moving train have constant speed, regardless of the train's speed. However, the waves will "bunch up" ahead of the train, causing a higher tone, and they will "lag" behind the train, causing a lower tone. This is the DOPPLER EFFECT for propagated waves.