To model a vibrating string using gnuplot and then ffmpeg.

string_vibrate.c

#include <stdio.h>
#include <stdint.h>
#include <stdlib.h>
#include <math.h>

#define TIME_OF_VIDEO     (uint32_t) 10
#define LENGTH_OF_STRING  (uint32_t) 3
#define SAMPLE_PER_LENGTH (uint32_t) 50
#define FPS               (uint32_t) 30
#define FREQUENCY         (uint32_t) 1
#define MODE              (uint32_t) 3

double my_func(double t, double x){
   return (double) (3*sin(2*M_PI*FREQUENCY*t)*sin(x*MODE*M_PI/LENGTH_OF_STRING));
}

int main(){
   FILE *fp = NULL;
   char command[300];

   double length_period = (double) 1/SAMPLE_PER_LENGTH;
   double fps_period = (double) 1/FPS;

   for (uint32_t frame = 0; frame < TIME_OF_VIDEO*FPS; frame++){
      fp = fopen("pl.dat", "w");

      if (!fp) return -1; // check with `echo $?`

      for (uint32_t x = 0; x < LENGTH_OF_STRING*SAMPLE_PER_LENGTH; x++){

         fprintf(fp, "%.3f  %.3f\n", (double) x*length_period, my_func((double)frame*fps_period, (double) x));
      }

      fclose(fp);
      sprintf(command, "gnuplot -e \"set term png; set xlabel \'plot-%d.png\'; plot [0:%d] [-3:3] \'pl.dat\' with lines\" > plot-%03d", frame, LENGTH_OF_STRING. frame);                      >
      printf("%s\n", command);
      system(command);

   }
   return 0; // check with `echo $?`

}