### 6.4 Waves in Space

With the function plotter described above waves in space can be presented quite vividly. Then one or more space variable appear in a periodic function , for example as $Math content$. The spatial surface will then be periodic in or two dimensions. In the simulation for Fig.6.9 a number of such waves is preset.

We daily observe surface waves in a multitudes of shapes on water. In general these waves propagate in time in one direction without changing their character noticeably in small regions of space. In the simulation this can be reproduced by adding a phase $Math content$ and incrementing the time $Math content$ continuously and evenly: $Math content$. The wave that is stationary for $Math content$ moves for $Math content$ with constant velocity in positive $Math content$-direction. The propagation velocity is set with $Math content$. This animation makes the projection picture of the wave very vivid.

The following functions are preset in the selection field.

 functions plane wave in $Math content$ plane wave in $Math content$ plane wave with arbitrary direction concurrent interference $Math content$ opposing interference $Math content$ concurrent interference $Math content$ +$Math content$ concurrent interference $Math content$ +$Math content$ orthogonal interference $Math content$ + $Math content$ concurrent interference, adjustable angle $Math content$ opposing interference, adjustable angle $Math content$ diverging radial wave converging radial wave stationary radial wave diverging surface wave diverging space wave formula in simulation syntax a*sin(b*x-p*t) a*sin(b*y-p*t): 0.3*sin(6*pi*a*(b*y+c*x)/sqrt(b*b+c*c)-p*t) a*(sin(b*y-p*t)+sin(b*y-p*t)) a*(sin(b*y-p*t)+sin(-b*y-p*t)) a*(sin(b*y-p*t)+sin(c*y-p*t)) a*(sin(b*y-p*t)+sin(-c*y-p*t)) a*(sin(b*x-p*t)+sin(c*y-p*t)) a*(sin(b*(y-(c-pi)*x)-p*t)+sin(b*(y+(c-pi)*x)-p*t)) a*(sin(b*(y-(c-pi)*x)-p*t)+sin(b*(-y+(c-pi)*x)-p*t)) a*sin(b*(x*x+y*y)-p*t) a*sin(b*(x*x+y*y)+p*t) a*(sin(b*(xˆ2+yˆ2)-p*t)+sin(b*(xˆ2+yˆ2)+p*t)) 0.4*a*sin(b*(xˆ2+yˆ2)-p*t)/sqrt(0.1+xˆ2+yˆ2) 0.2*a*sin(b*(xˆ2+yˆ2)-p*t)/(0.1+xˆ2+yˆ2)

The interference of waves with the same direction of propagation is referred to as concurrent interference. that of opposite direction as opposing interference. We also give examples for the interference of waves of the same frequency as well as of waves of different frequency and finally the interference of waves under 90 degree and under adjustable angles.

For the radial waves the simple radial wave with constant amplitude is physically not possible, i.e. it is a unrealistic fiction. This is because the amplitude will decay as function of the radius (distance from the excitation center), since the excitation energy is distributed over a larger and larger circle. For the spatial radial wave, for example the spatial compression wave originating from a nearly point-like source, the section of the excitation is shown in the $Math content$-plane; here the amplitude decays with the radius, i.e. like $Math content$. since the energy is distributed over a spherical surface

With this simulation you can train your spatial awareness for wave phenomena and the corresponding understanding of formulas. When editing the formulas you can explore many possibilities to simulate natural phenomena. Remember, that you may also choose the velocity of propagation differently when superimposing several waves and thus observe the phenomenon of dispersion. Further instructions can be found in the description pages.

These animations start in a state of motion. You also may change parameters while the animation is running and switch between function types. Fig6.9 shows as example a radial wave in space.