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Both sides previous revision Previous revision Next revision | Previous revision Next revision Both sides next revision | ||
cs:hydrophones:trilateration_setup:start [2016/11/16 16:45] Ryan Summers |
cs:hydrophones:trilateration_setup:start [2016/12/14 00:49] Brian Moore |
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\Delta z = \Delta t_z * c_s \\ | \Delta z = \Delta t_z * c_s \\ | ||
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- | In other words, $h_x$ is $\Delta x$ meters farther from the pinger than $h_0$, and $h_0$ is $p_0$ meters from the pinger. | + | In other words, $h_x$ is $\Delta x$ meters closer the pinger than $h_0$, and $h_0$ is $p_0$ meters from the pinger. |
The final calculations for $x$, $y$, and $z$ will be in terms of $\Delta x$, $\Delta y$, and $\Delta z$ | The final calculations for $x$, $y$, and $z$ will be in terms of $\Delta x$, $\Delta y$, and $\Delta z$ | ||
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x^2 = p_0^2 \frac{4\Delta x^2}{4 \delta^2} + p_0 \frac{4\Delta x}{4\delta^2}(-\Delta x^2 + \delta^2) + \frac{(\Delta x^2 - \delta^2)^2}{4 \delta^2} \\ | x^2 = p_0^2 \frac{4\Delta x^2}{4 \delta^2} + p_0 \frac{4\Delta x}{4\delta^2}(-\Delta x^2 + \delta^2) + \frac{(\Delta x^2 - \delta^2)^2}{4 \delta^2} \\ | ||
- | x^2 = p_0^2 \left(\frac{\Delta x}{\delta}\right)^2 + p_0 \frac{\Delta x}{\delta^2}(\delta^2 +\Delta x^2) + \left(\frac{\Delta x^2 - \delta^2}{2 \delta}\right)^2 \\ | + | x^2 = p_0^2 \left(\frac{\Delta x}{\delta}\right)^2 + p_0 \frac{\Delta x}{\delta^2}(\delta^2 -\Delta x^2) + \left(\frac{\Delta x^2 - \delta^2}{2 \delta}\right)^2 \\ |
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We can derive a similar equation for $y^2$ and $z^2$ | We can derive a similar equation for $y^2$ and $z^2$ | ||
$$ | $$ | ||
- | x^2 = p_0^2 \left(\frac{\Delta x}{\delta}\right)^2 + p_0 \frac{\Delta x}{\delta^2}(\delta^2 +\Delta x^2) + \left(\frac{\Delta x^2 - \delta^2}{2 \delta}\right)^2 \\ | + | x^2 = p_0^2 \left(\frac{\Delta x}{\delta}\right)^2 + p_0 \frac{\Delta x}{\delta^2}(\delta^2 -\Delta x^2) + \left(\frac{\Delta x^2 - \delta^2}{2 \delta}\right)^2 \\ |
- | y^2 = p_0^2 \left(\frac{\Delta y}{\epsilon}\right)^2 + p_0 \frac{\Delta y}{\epsilon^2}(\epsilon^2 +\Delta y^2) + \left(\frac{\Delta y^2 - \epsilon^2}{2 \epsilon}\right)^2 \\ | + | y^2 = p_0^2 \left(\frac{\Delta y}{\epsilon}\right)^2 + p_0 \frac{\Delta y}{\epsilon^2}(\epsilon^2 -\Delta y^2) + \left(\frac{\Delta y^2 - \epsilon^2}{2 \epsilon}\right)^2 \\ |
- | z^2 = p_0^2 \left(\frac{\Delta z}{\zeta}\right)^2 + p_0 \frac{\Delta z}{\zeta^2}(\zeta^2 +\Delta z^2) + \left(\frac{\Delta z^2 - \zeta^2}{2 \zeta}\right)^2 \\ | + | z^2 = p_0^2 \left(\frac{\Delta z}{\zeta}\right)^2 + p_0 \frac{\Delta z}{\zeta^2}(\zeta^2 -\Delta z^2) + \left(\frac{\Delta z^2 - \zeta^2}{2 \zeta}\right)^2 \\ |
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a_x = \left(\frac{\Delta x}{\delta}\right)^2 \qquad | a_x = \left(\frac{\Delta x}{\delta}\right)^2 \qquad | ||
- | b_x = \frac{\Delta x}{\delta^2}(\delta^2 +\Delta x^2) \qquad | + | b_x = \frac{\Delta x}{\delta^2}(\delta^2 -\Delta x^2) \qquad |
c_x = \left(\frac{\Delta x^2 - \delta^2}{2 \delta}\right)^2 \\ | c_x = \left(\frac{\Delta x^2 - \delta^2}{2 \delta}\right)^2 \\ | ||
a_y = \left(\frac{\Delta y}{\epsilon}\right)^2 \qquad | a_y = \left(\frac{\Delta y}{\epsilon}\right)^2 \qquad | ||
- | b_y = \frac{\Delta y}{\epsilon^2}(\epsilon^2 +\Delta y^2) \qquad | + | b_y = \frac{\Delta y}{\epsilon^2}(\epsilon^2 -\Delta y^2) \qquad |
c_y = \left(\frac{\Delta y^2 - \epsilon^2}{2 \epsilon}\right)^2 \\ | c_y = \left(\frac{\Delta y^2 - \epsilon^2}{2 \epsilon}\right)^2 \\ | ||
a_z = \left(\frac{\Delta z}{\zeta}\right)^2 \qquad | a_z = \left(\frac{\Delta z}{\zeta}\right)^2 \qquad | ||
- | b_z = \frac{\Delta z}{\zeta^2}(\zeta^2 +\Delta z^2) \qquad | + | b_z = \frac{\Delta z}{\zeta^2}(\zeta^2 -\Delta z^2) \qquad |
c_z = \left(\frac{\Delta z^2 - \zeta^2}{2 \zeta}\right)^2 | c_z = \left(\frac{\Delta z^2 - \zeta^2}{2 \zeta}\right)^2 | ||
$$ | $$ |