Generating (and Saving) Tones with SoX

I’ve been using the command line program SoX to generate tones for my physics demonstrations on sound waves.

Single frequency tones can be used for talking about frequency and wavelength, as well as discussing octaves.

Combine two tones allows you to talk about interference and beats.

SoX can do a lot more than this, so I though I’d compile what I’m using it for in a single, reference post. For the record: I’m using SoX in Terminal on a Mac.

Using SoX

To play a single note (frequency 173.5 Hz) for 5 seconds, use:

> play -n  synth 5 sin 347

To save the note to a mp3 file (called note.mp3) use:

> sox -n note.mp3 synth 5 sin 347

The SoX command to play two notes with frequencies of 347 and 357 Hz is:

> play -n synth 15 sin 347 sin 357

and to make an mp3 file use:

> sox -n beat_10.mp3 synth 15 sin 347 sin 357

Everything You (N)ever Wanted to Know About Parabolas

So that my students could more easily check their answers graphically, I put together a page with a more complete analysis of parabolas (click this link for more details).

[inline]

Analyzing Parabolas

Standard Form	Vertex Form
`y = a x² + b x + c`	`y = a (x – h)² + k`
y = x² + x +	y = ( x – ) ² +

Intercepts:	Vertex:
Focus:	Directrix:	Axis:

Solution by Factoring:

y = x² x

[script type=”text/javascript”]
var width=500;
var height=500;
var xrange=10;
var yrange=10;

mx = width/(2.0*xrange);
bx = width/2.0;
my = -height/(2.0*yrange);
by = height/2.0;

function draw_9239(ctx, polys) {
t_9239=t_9239+dt_9239;
//ctx.fillText (“t=”+t, xp(5), yp(5));
ctx.clearRect(0,0,width,height);

polys[0].drawAxes(ctx);
ctx.lineWidth=2;
polys[0].draw(ctx);

polys[0].write_eqn2(ctx);

//polys[0].y_intercepts(ctx);
//write intercepts on graph

// SHOW VERTEX
if (show_vertex_ctrl_9239.checked==true) {
polys[0].draw_vertex(ctx);
document.getElementById(‘vertex_pos_9239’).innerHTML = “(“+polys_9239[0].vertex.x.toPrecision(2)+ ” , “+polys_9239[0].vertex.y.toPrecision(2)+ ” )”;
} else { document.getElementById(‘vertex_pos_9239’).innerHTML = “”;}

// SHOW FOCUS
if (show_focus_ctrl_9239.checked==true) {
polys[0].draw_focus(ctx);
document.getElementById(‘focus_pos_9239’).innerHTML = “(“+polys_9239[0].focus.x.toPrecision(2)+ ” , “+polys_9239[0].focus.y.toPrecision(2)+ ” )”;
} else { document.getElementById(‘focus_pos_9239’).innerHTML = “”;}

// SHOW DIRECTRIX
if (show_directrix_ctrl_9239.checked==true) {
polys[0].draw_directrix(ctx);
document.getElementById(‘directrix_pos_9239’).innerHTML = polys[0].directrix.get_eqn2(“y”,”x”,”html”);

polys[0].directrix.write_eqn2(ctx, polys[0].directrix.get_eqn2(“directrix: y”));
} else { document.getElementById(‘directrix_pos_9239’).innerHTML = “”;}

// SHOW AXIS
if (show_axis_ctrl_9239.checked==true) {
polys[0].draw_parabola_axis(ctx);
document.getElementById(‘axis_pos_9239’).innerHTML = “x = “+polys[0].vertex.x.toPrecision(2);
}

//SHOW INTERCEPTS
ctx.textAlign=”center”;
if (show_intercepts_ctrl_9239.checked==true) {
polys[0].x_intercepts(ctx);
ctx.fillText (‘x intercepts: (when y=0)’, xp(6), yp(8));
//ctx.fillText (‘intercepts=’+polys[0].x_intcpts.length, xp(5), yp(-5));
if (polys[0].order == 2) {
if (polys[0].x_intcpts.length > 0) {
line = “0 = “;
for (var i=0; i 0.0) { sign=”-“;} else {sign=”+”;}
line = line + “(x “+sign+” “+ Math.abs(polys[0].x_intcpts[i].toPrecision(2))+ “)”;
}
ctx.fillText (line, xp(6), yp(7));
for (var i=0; i 0) {
for (var i=0; i2 “+polys[0].bsign+” “+Math.abs(polys[0].b.toPrecision(2))+” x “+ polys[0].csign+” “+Math.abs(polys[0].c.toPrecision(2))+”

“;

solution = solution + ‘Factoring: ‘;

if (polys[0].x_intcpts.length > 0) {
solution = solution + ‘0 = ‘;
for (var i=0; i 0.0) { sign=”-“;} else {sign=”+”;}
solution = solution + “(x “+sign+” “+ Math.abs(polys[0].x_intcpts[i].toPrecision(2))+ “)”;
}
solution = solution + ‘

‘;
solution = solution + ‘Set each factor equal to zero:
‘;
for (var i=0; i 0.0) { sign=”-“;} else {sign=”+”;}
solution = solution + “x “+sign+” “+ Math.abs(polys[0].x_intcpts[i].toPrecision(2))+ ” = 0 “;
}
solution = solution + ‘

and solve for x:
‘;
for (var i=0; i‘;
}
document.getElementById(‘equation_9239’).innerHTML = solution;
}

else if (polys[0].order == 1) {
solution = solution + ‘
‘;
solution = solution + “y = “+” “+Math.abs(polys[0].b.toPrecision(2))+” x “+ polys[0].csign+” “+Math.abs(polys[0].c.toPrecision(2))+”

“;
solution = solution + ‘

Set y=0 and solve for x:
‘;
solution = solution + ” 0 = “+” “+Math.abs(polys[0].b.toPrecision(2))+” x “+ polys[0].csign+” “+Math.abs(polys[0].c.toPrecision(2))+”

“;
solution = solution + ‘ ‘;
solution = solution + (-1.0*polys[0].c).toPrecision(2) +” = “+” “+Math.abs(polys[0].b.toPrecision(2))+” x “+”

“;
solution = solution + ‘ ‘;
solution = solution + (-1.0*polys[0].c).toPrecision(2)+”/”+polys[0].b.toPrecision(2)+” = “+” x “+”

“;
solution = solution + ‘ ‘;
solution = solution + “x = “+ (-1.0*polys[0].c/polys[0].b).toPrecision(4)+”

“;

document.getElementById(‘equation_9239’).innerHTML = solution;
}

}

function update_form_9239 () {
a_coeff_9239.value = polys_9239[0].a+””;
b_coeff_9239.value = polys_9239[0].b+””;
c_coeff_9239.value = polys_9239[0].c+””;

av_coeff_9239.value = polys_9239[0].a+””;
hv_coeff_9239.value = polys_9239[0].h+””;
kv_coeff_9239.value = polys_9239[0].k+””;

}

//init_mouse();

var c_9239=document.getElementById(“myCanvas_9239”);
var ctx_9239=c_9239.getContext(“2d”);

var change = 0.0001;

function create_lines_9239 () {
//draw line
//document.write(“hello world! “);
var polys = [];
polys.push(addPoly(1,6, 5));

// polys.push(addPoly(0.25, 1, 0));
// polys[1].color = ‘#8C8’;

return polys;
}

var polys_9239 = create_lines_9239();

var x1=xp(-10);
var y1=yp(1);
var x2=xp(10);
var y2=yp(1);
var dc_9239=0.05;

var t_9239 = 0;
var dt_9239 = 100;
//end_ct = 0;
var st_pt_x_9239 = 2;
var st_pt_y_9239 = 1;
var show_vertex_9239 = 1; //1 to show vertex on startup
var show_focus_9239 = 1; // 1 to show the focus
var show_intercepts_9239 = 1; // 1 to show the intercepts
var show_directrix_9239 = 1; // 1 to show the directrix
var show_axis_9239 = 1; //1 to show the axis of the parabola

var move_dir_9239 = 1.0; // 1 for up

//document.getElementById(‘comment_spot’).innerHTML = polys_9239[0].a+” “+polys_9239[0].b+” “+polys_9239[0].c+” : “+polys_9239[0].h+” “+polys_9239[0].k+” “;

//standard form
var a_coeff_9239 = document.getElementById(“a_coeff_9239”);
var b_coeff_9239 = document.getElementById(“b_coeff_9239”);
var c_coeff_9239 = document.getElementById(“c_coeff_9239”);

//vertex form
var av_coeff_9239 = document.getElementById(“av_coeff_9239”);
var hv_coeff_9239 = document.getElementById(“hv_coeff_9239”);
var kv_coeff_9239 = document.getElementById(“kv_coeff_9239”);

//options
var show_vertex_ctrl_9239 = document.getElementById(“show_vertex_9239”);
if (show_vertex_9239 == 0) {show_vertex_ctrl_9239.checked=false;
} else {show_vertex_ctrl_9239.checked=true;}

var show_focus_ctrl_9239 = document.getElementById(“show_focus_9239”);
if (show_focus_9239 == 0) {show_focus_ctrl_9239.checked=false;
} else {show_focus_ctrl_9239.checked=true;}

var show_intercepts_ctrl_9239 = document.getElementById(“show_intercepts_9239”);
if (show_intercepts_9239 == 0) {show_intercepts_ctrl_9239.checked=false;
} else {show_intercepts_ctrl_9239.checked=true;}

var show_directrix_ctrl_9239 = document.getElementById(“show_directrix_9239”);
if (show_directrix_9239 == 0) {show_directrix_ctrl_9239.checked=false;
} else {show_directrix_ctrl_9239.checked=true;}

var show_axis_ctrl_9239 = document.getElementById(“show_axis_9239”);
if (show_axis_9239 == 0) {show_axis_ctrl_9239.checked=false;
} else {show_axis_ctrl_9239.checked=true;}

update_form_9239();

//document.write(“test= “+c_coeff_9239.value+” “+polys_9239[0].c);
setInterval(“draw_9239(ctx_9239, polys_9239)”, dt_9239);

a_coeff_9239.onchange = function() {
//polys_9239[0].a = parseFloat(this.value);
polys_9239[0].set_a(parseFloat(this.value));
polys_9239[0].update_vertex_form_parabola();
update_form_9239();
}
b_coeff_9239.onchange = function() {
//polys_9239[0].b = parseFloat(this.value);
polys_9239[0].set_b(parseFloat(this.value));
polys_9239[0].update_vertex_form_parabola();
update_form_9239();
}
c_coeff_9239.onchange = function() {
//polys_9239[0].c = parseFloat(this.value);
polys_9239[0].set_c(parseFloat(this.value));
polys_9239[0].update_vertex_form_parabola();
update_form_9239();
}

av_coeff_9239.onchange = function() {
//polys_9239[0].a = parseFloat(this.value);
polys_9239[0].set_a(parseFloat(this.value));
polys_9239[0].update_standard_form_parabola();
update_form_9239();
}

hv_coeff_9239.onchange = function() {
polys_9239[0].h = parseFloat(this.value);
polys_9239[0].update_standard_form_parabola();
polys_9239[0].set_order()
update_form_9239();
}

kv_coeff_9239.onchange = function() {
polys_9239[0].k = parseFloat(this.value);
polys_9239[0].update_standard_form_parabola();
polys_9239[0].set_order()
update_form_9239();
}

//draw_9239();
//document.write(“x”+x2+”x”);
//ctx_9239.fillText (“n=”, xp(5), yp(5));

[/script]

[/inline]

Converting the forms

The key relationships are the ones to convert from the standard form of the parabolic equation:

$y = a x^2 + b x + c$

(1)

to the vertex form:

$y = a (x - h)^2 + k$

(2)

If you multiply out the vertex equation form you get:

y = a x² – 2ah x + ah² + k

(3)

When you compare this equation to the standard form of the equation (Equation 1), if you look at the coefficients and the constants, you can see that:

To convert from the vertex to the standard form use:

         $a = a$ (4)

         $b = -2ah$ (5)

         $c = ah^2 + k$ (6)

Going the other way,

To convert from the standard to the vertex form of parabolic equations use:

         $a = a$
(7)

         $h = \frac{-b}{2a}$
(8)

         $k = c - ah^2$
(9)

Although it is sometimes convenient to let k not depend on coefficients from its own equation:

$k = c - \frac{b^2}{4a}$ (10)

The Vertex and the Axis

The nice thing about the vertex form of the equation of the parabola is that if you want the find the coordinates of the vertex of the parabola, they’re right there in the equation.

Specifically, the vertex is located at the point:

$(x_v, y_v) = (h, k)$

(11)

The axis of the parabola is the vertical line going through the vertex, so:

The equation for the axis of a parabola is:

$x = h$

(12)

Focus and Directrix

Finally, it’s important to note that the distance (d) from the vertex of the parabola to its focus is given by:

$d = \frac{1}{4a}$

(13)

Which you can just add d on to the coordinates of the vertex (Equation 11) to get the location of the focus.

$(x_f, y_f) = (x_v, y_v + d)$

(14)

The directrix is just the opposite, vertical distance away, so the equation for the directrix is the equation of the horizontal line at:

$y = y_v + d$

(15)

References

There are already some excellent parabola references out there including:

Wonderful interactive graphs by Murray Bourne.
Nice simple explanations of the different forms of the equations.
More complex explanations from Wolfram Alpha.

Euler’s Method for Approximating a Solution to a Differential Equation

Based on Euler’s Method, this interactive graph illustrates a numerical method for solving differential equations. This approach is at the core of many sophisticated computer models of physical phenomena (like climate and weather).

[inline]

Starting point: (x,y) = ( , )
Step size:
Direction:

Slope equation: dy/dx = x +
Show analytical solution:

If you know the equation for the slope of a curve (the red line for example),

and a point that the curve passes through, such as , you can integrate to find the equation of the curve:

[script type=”text/javascript”]

var width=500;
var height=500;
var xrange=10;
var yrange=10;

mx = width/(2.0*xrange);
bx = width/2.0;
my = -height/(2.0*yrange);
by = height/2.0;

function upause () {
ct = 1;
}

function draw_9110(ctx, polys) {
t_9110=t_9110+dt_9110;
//ctx.fillText (“t=”+t, xp(5), yp(5));
ctx.clearRect(0,0,width,height);

polys[0].drawAxes(ctx);
polys[0].slopeField(ctx, 1.0, 1.0, 0.5);
ctx.lineWidth=2;
polys[0].draw(ctx);
polys[0].write_eqn(ctx, “slope = dy/dx”);

eu = eu_9110;
//override nsteps values
//max_nsteps_9110 = eu.nsteps;
//eu.nsteps = nsteps_9110;

if (nsteps_9110 <= max_nsteps_9110) { nsteps_9110 = nsteps_9110 + 1;} else { nsteps_9110 = 0; } eu.nsteps = nsteps_9110; if (show_parabola_9110 == 1) { polys[0].Integrate(ctx, eu.x1, eu.y1); polys[0].integral.color = '#3c3'; polys[0].integral.draw(ctx); } pos = polys[0].EulerApprox(ctx, eu); //document.getElementById('comment_9110').innerHTML = eu.nsteps+ " " + eu.x1 + " " + eu.y1+ " " + eu.dx+ " " + eu.dir+ " " + max_nsteps_9110; //starting point label ctx.textAlign="right"; ctx.textBaseline="alphabetic"; ctx.font = "14pt Calibri"; ctx.fillStyle = "#00f"; ctx.fillText ('start = ('+eu.x1+','+eu.y1+")", xp(eu.x1-0.25), yp(eu.y1-0.5)); //pos = polys[0].EulerApprox(ctx, st_pt_x_9110, st_pt_y_9110, 0.5,20,-1); //pos = polys[0].EulerApprox(ctx, st_pt_x_9110, st_pt_y_9110, 1, 6,1); //draw starting point ctx.strokeStyle="#00f"; ctx.lineWidth=2; ctx.beginPath(); ctx.arc(xp(eu.x1),yp(eu.y1),dxp(0.1),0,Math.PI*2); ctx.closePath(); ctx.stroke(); //ctx.fillText (' c='+move_dir_9110+' '+polys[1].c, xp(5), yp(5)); } //init_mouse(); var c_9110=document.getElementById("myCanvas_9110"); var ctx_9110=c_9110.getContext("2d"); var change = 0.0001; function create_lines_9110 () { //draw line var polys = []; polys.push(addPoly(0, 0.5, -1)); return polys; } function set_max_nsteps_9110 (eu) { max_nsteps_9110 = 1.5*(xrange - eu.x1)/eu.dx; } var polys_9110 = create_lines_9110(); var x1=xp(-10); var y1=yp(1); var x2=xp(10); var y2=yp(1); var dc_9110=0.05; var t_9110 = 0; var dt_9110 = 500; var st_dx_9110 = 1.0; var st_nsteps_9110 = 0; var nsteps_9110 = 0; var max_nsteps_9110 = 20; var dir_9110 = 1; var show_parabola_9110 = 1; var st_pt_x_9110 = -2; var st_pt_y_9110 = -3; var move_dir_9110 = 1.0; // 1 for up //Form interaction function update_form_9110 (x, y, dx, nsteps, dir, b, c, show_para) { x_9110.value = st_pt_x_9110+""; y_9110.value = st_pt_y_9110+""; dx_9110.value = st_dx_9110.toPrecision(2); //c_max_nsteps_9110.value = max_nsteps_9110+""; //document.getElementById('comment_9110').innerHTML = c_dir_9110; if (dir == 1) { c_dir_9110.selectedIndex = 0; } else {c_dir_9110.selectedIndex = 1;} b_9110.value = b+""; c_9110.value = c+""; if (show_para == 1) { ctrl_show.checked = 1; } else {ctrl_show.checked = 0;} } function get_euler_form_9110 () { pos = {x : parseFloat(x_9110.value), y : parseFloat(y_9110.value)} if (c_dir_9110.selectedIndex == 0) { dir = 1;} else {dir = -1;} eu = addEuler(pos.x, pos.y, parseFloat(dx_9110.value), max_nsteps_9110, dir); return eu } function update_solution_9110 (ctx, eu) { //update equations and solution document.getElementById('slope_eqn_9110').innerHTML = polys_9110[0].get_eqn("dy/dx"); //document.getElementById('eqn_9110').innerHTML = "hey"+eu.y1; polys_9110[0].Integrate(ctx, eu.x1, eu.y1); document.getElementById('eqn_9110').innerHTML = polys_9110[0].integral.get_eqn(); document.getElementById('init_pt_9110').innerHTML = "("+eu.x1+","+eu.y1+")"; } function add_euler_line() { eu_9110 = get_euler_form_9110 (); max_nsteps_9110 = eu_9110.nsteps; nsteps_9110 = 0; } var x_9110 = document.getElementById("x_9110"); var y_9110 = document.getElementById("y_9110"); var dx_9110 = document.getElementById("dx_9110"); //var c_max_nsteps_9110 = document.getElementById("nsteps_9110"); var c_dir_9110 = document.getElementById("dir_9110"); var b_9110 = document.getElementById("b_9110"); var c_9110 = document.getElementById("c_9110"); var ctrl_show = document.getElementById("chk_show_para_9110"); update_form_9110(x_9110, y_9110, dx_9110, nsteps_9110, dir_9110, polys_9110[0].b, polys_9110[0].c, show_parabola_9110); var eu_9110 = get_euler_form_9110 (); set_max_nsteps_9110(eu_9110); update_solution_9110(ctx_9110, eu_9110); setInterval("draw_9110(ctx_9110, polys_9110)", dt_9110); x_9110.onchange = function() { eu_9110 = get_euler_form_9110 (); max_nsteps_9110 = eu_9110.nsteps; nsteps_9110 = 0; } y_9110.onchange = function() { eu_9110 = get_euler_form_9110 (); max_nsteps_9110 = eu_9110.nsteps; nsteps_9110 = 0; } c_dir_9110.onchange = function() { eu_9110 = get_euler_form_9110 (); max_nsteps_9110 = eu_9110.nsteps; nsteps_9110 = 0; } dx_9110.onchange = function() { eu_9110 = get_euler_form_9110 (); max_nsteps_9110 = eu_9110.nsteps; //max_nsteps_9110 = 1.4 * (10 - eu_9110.x1)/eu_9110.dx; set_max_nsteps_9110(eu_9110); nsteps_9110 = 0; } b_9110.onchange = function() { polys_9110[0].set_b(parseFloat(this.value)); eu_9110 = get_euler_form_9110 (); max_nsteps_9110 = eu_9110.nsteps; nsteps_9110 = 0; } c_9110.onchange = function() { polys_9110[0].set_c(parseFloat(this.value)); eu_9110 = get_euler_form_9110 (); max_nsteps_9110 = eu_9110.nsteps; nsteps_9110 = 0; } ctrl_show.onchange = function() { if (this.checked == 1) { show_parabola_9110 = 1;} else {show_parabola_9110 = 0} } [/script] [/inline]

If you don’t have a starting point (initial condition), you can draw a slope field to see what the general pattern of all the possible solutions.

Even with a starting point, however, there are just times when you can’t integrate the slope equation — it’s either too difficult or even impossible.

Then, what you can do is come up with an approximation of what the curve looks like by projecting along the slope from the starting point.

The program above demonstrates how it’s done. This approach is called Euler’s Method, and is gives a numerical approximation rather than finding the exact, analytical solution using calculus (integration).

So why use an approximation when you can find the exact solution? Because, there are quite a number of problems that are impossible or extremely difficult to solve analytically, things like: the diffusion of pollution in a lake; how changing temperature in the atmosphere gives you weather and climate; the flow of groundwater in aquifers; stresses on structural members of buildings; and the list goes on and on.

As with most types of numerical approximations, you get better results if you can reduce the step size between projections of the slope. Try changing the numbers and see.

A more detailed version, with solutions, is here: Euler’s Method.

A good reference: Euler’s Method by Paul Dawkins.

Test HTML5 Canvas

[inline]

[script type=”text/javascript”]
document.write(“hello world!”);

var c2=document.getElementById(“myCanvas2”);
var ctx2=c2.getContext(“2d”);
ctx2.moveTo(10,10);
ctx2.lineTo(150,50);
ctx2.lineTo(10,50);
ctx2.stroke();
[/script]
[/inline]

So I’m experimenting with creating drawings using HTML5. And it works (mostly)!! I have to hardwire in paragraph breaks, but otherwise it works so far.

The key references:

w3schools.com: HTML5 Canvas: for the code for the drawing above.
Volcano’s Inline Javascript plugin for WordPress to make it work on this blog.
The page for Opera developers is also a great HTML5 drawing reference.
Dive into HTML5 is another excellent resource.
Bill Mill’s canvas tutorial.

Tess of the d’Urbervilles: Police Composite

Brian Joseph Davis creates composite sketches of literary characters using the same software used by the police, and the descriptions of the characters in the books.

tess-durbaville — Composite sketch of Tess from Thomas Hardy's novel Tess of the d'Urbervilles. Image by Brian Joseph Davis.

She was a fine and handsome girl—not handsomer than some others, possibly—but her mobile peony mouth and large innocent eyes added eloquence to colour and shape… The pouted-up deep red mouth to which this syllable was native had hardly as yet settled into its definite shape, and her lower lip had a way of thrusting the middle of her top one upward, when they closed together after a word…Phases of her childhood lurked in her aspect still. As she walked along to-day, for all her bouncing handsome womanliness, you could sometimes see her twelfth year in her cheeks, or her ninth sparkling from her eyes…a thick cable of twisted dark hair hanging straight down her back to her waist.

— description by Thomas Hardy, excerpted by Davis (2010): The Composites.

Wikipedia’s Green Skies

“If all historians save one say that the sky was green in 1888, our policies require that we write ‘Most historians write that the sky was green, but one says the sky was blue.’ … As individual editors, we’re not in the business of weighing claims, just reporting what reliable sources write.”

— Wikipedia editor quoted in Messer-Kruse (2012): The ‘Undue Weight’ of Truth on Wikipedia in The Chronicle of Higher Education.

You might be an expert on a subject, and you may have the facts and primary documents to back you up, but if the majority of sources (who are clearly wrong) disagree with you then the Wikipedia policy is to go with the majority.

For an openly edited encyclopedia this policy actually makes a lot of sense. Otherwise, anyone who calls themself an expert would be able to install their point of view. The downside, however, is that Wikipedia will trail current research by a significant margin.

It’s a check against the citogenesis process:

Food for though when tempted to Wikipedia as a reference.

How Google Works

One of the librarians at Washington University played this video for our students. It was a great supplement to their lesson on how to use the internet for research.