OK, here's my suggestion.
There are three parts to the design: two versions of the transmitter, and the receiver.
I strongly recommend you use the crystal-controlled transmitter because it requires no setup and will not drift, but it's more complicated than the 555-based transmitter.
If you use the 555-based transmitter, you will need to adjust the frequency. If you have an oscilloscope, a frequency counter, or a multimeter that shows frequency, you can use that to adjust it for 40 kHz, otherwise you will have adjust it for best detection range.
All circuit sections require a power supply in the range 9~12V DC.
Crystal-controlled transmitter (component references starting with 1):
1U1 forms a standard Pierce oscillator with crystal 1X2, producing a 4.0 MHz signal at 1U1 pin 12 which is used to clock 1U2, a binary counter/divider. Outputs Q2, Q5 and Q6 are diode-ANDed together to produce a signal at RST that goes high when the count reaches 110010 binary (50 decimal). This creates an 80 kHz signal at Q6 which is then divided by two in 1U3 to produce a 40 kHz signal with exactly 50% duty cycle.
This signal is buffered by 1Q1 and 1Q2 operating as complementary emitter followers, and the output drives ultrasonic transmitter 1X1. 1C6 and 1R4 ensure that the mean DC voltage across 1X1 is 0V.
The three decoupling capacitors, 1C3, 1C4 and 1C5, must be connected as directly as possible between the VDD and VSS pins of their respective ICs.
555-based transmitter (component references starting with 2):
2U1 is a 555 used in the standard configuration. The operating frequency is determined by 2R1, 2R2, 2R3 and 2C1 according to the standard formula. The use of a relatively low value for 2R2 means that the output signal has a duty cycle fairly close to 50%.
2C1 must be a good-quality, low-tolerance unit; a suitable Digi-Key part number is given on the schematic.
The output is buffered by 2Q1 and 2Q2 and coupled to ultrasonic transmitter 2X1.
Receiver (component references starting with 3):
3R1 and 3R2 create a half-supply-voltage point for op-amp bias. 3R3 keeps the mean DC voltage across the ultrasonic receiver, 3X1, around 0V, and 3D1 and 3D2 protect op-amp 3U1A from voltage transients. 3X1 should be connected to the board using a short length of screened cable; the screen and core conductor are labelled on the diagram.
3U1A provides a fixed gain of 47 (3R5 / 3R4). 3U1B provides additional gain, according to 3R7 / 3R6. 3R7 must be selected to give the desired sensitivity. Increasing the value increases the sensitivity. Start with a value of 10k and adjust accordingly.
The detected 40 kHz signal at 3U1B's output is passed through a charge pump consisting of 3C3, 3D3, 3D4 and 3C4 where the level is converted into a DC voltage (with a loss of about 1.3V due to diode drops).
The voltage across 3C4 is passed into a transistor Schmitt trigger, to provide clean switching of the relay. With the values given, the relay will turn ON when the 3C4 voltage reaches about 1.5V and it will turn back OFF again when the 3C4 voltage drops below about 0.7V.
Relay 3K1 must have a coil voltage equal to the supply voltage. It will activate when the received ultrasonic signal is greater than the detection threshold (which is determined by the amount of gain in the second op-amp stage, set by 3R7).