How does a tranverse wave propagate?

Sound waves can be understood as particles hitting each other and to conserve momentum the vibration travels in air. Each particle transfering it's momentum to the other until it reaches our ears. Atleast we can think of a mental picture of why they propagate. But what about transverse waves? Like for instance when you jerk a rope or a slinky? Can somebody give me an intuitive reason for the propagation of these waves or better (if possible) a simple mathematical model?

• In case of mechanical transverse wave, it is the tension which is responsible for the wave propagation. Tension force from the neighbourhood for the transverse motion of each particle in the string. – user36790 Apr 18 '16 at 5:50

You pull a small piece of a rope up, and as that piece goes up it pulls the piece adjacent to it up and as that piece goes up... When you move your hand back to it's original position you're applying a force to the piece again and it pulls the adjacent piece down, etc... Model of displacement as a function of position and time: $y(x,t) = y_{max} sin(kx-\omega t)$ where $k=2\pi/\lambda$ and $\omega = 2\pi/T$ where $\lambda$ is wavelength and $\omega$ is angular frequency.
When the energy is somewhat orthogonal to the wave motion, you need two dimensions to represent it. The simplest case to model is probably a hypothetical 1-dimensional string held taunt by 2 points that are a fixed distance apart. We assume constant tension throughout the string and that, being 1-dimensional, has 0 diameter. If we assume that when hit hard enough that the string can start to vibrate and we assume it has some period, then by Fourier series it should have an summation of $\sin(x)$ and $\cos(x)$ with different amplitudes and frequencies to represent it. If we consider that the string is fixed between two points with a constant distance, we'll notice that gives us all possible terms - the distance divided by positive integers to determine the wavelength for only $\sin(x)$ terms in the summation (they're all $\sin(x)$ because we can assume they have a 0 amplitude at each of the points). That's why on a guitar if you put your finger at the 12th fret (halfway) to generate a harmonic it sounds like it's an octave higher (double the frequency/half the length). You could visualize the waves traversing back and forth, point to point. There are more complex variations, such as having something akin to air resistance to reduce the amplitude of the vibrations over time, modeling a 2-d plane vibrating, or along other shapes than having them fixed at either end, and you just have to set up equations to match the assumptions of your model to solve what happens in the model you set up over time.
• Use \sin in place of sin. – user36790 Apr 18 '16 at 5:47